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A rounded rectangle is the shape obtained by taking the convex hull of four equal circles of radius and placing their centers at the four corners of a rectangle with side lengths and .A filled rounded rectangle with (or ) is called a stadium.The rounded rectangle has perimeter(1)A filled rounded rectangle has area(2)For a rounded square with (center) length and width , the corner radius can be determined by measuring the generalized diameter from the edge of one rounded corner to the diagonally opposite corner. From the Pythagorean theorem, the diagonal distance from the rounded corner to the corner of the circumscribed square is(3)and the corner radius is related to the edge length of the square circumscribing the corner circle by(4)Combining gives(5)(6)

A lune is a plane figure bounded by two circular arcs of unequal radii, i.e., a crescent. (By contrast, a plane figure bounded by two circular arcs of equal radius is known as a lens.) For circles of radius and whose centers are separated by a distance , the area of the lune is given by(1)where(2)is the area of the triangle with side lengths , , and . The second of these can be obtained directly by subtracting the areas of the two half-lenses whose difference producing the colored region above.In each of the figures above, the area of the lune is equal to the area of the indicated triangle. Hippocrates of Chios squared the above left lune (Dunham 1990, pp. 19-20; Wells 1991, pp. 143-144), as well as two others, in the fifth century BC. Two more squarable lunes were found by T. Clausen in the 19th century (Shenitzer and Steprans 1994; Dunham 1990 attributes these discoveries to Euler in 1771). In the 20th century, N. G. Tschebatorew..

Half a circle. The area of a semicircle of radius is given by(1)(2)(3)The weighted mean of is(4)(5)The semicircle is the cross section of a hemispherefor any plane through the z-axis.The perimeter of the curved boundary is given by(6)With , this gives(7)The perimeter of the semicircular lamina is then(8)The weighted value of of the semicircular curve is given by(9)(10)(11)so the geometric centroid is(12)The geometric centroid of the semicircularlamina is given by(13)(Kern and Bland 1948, p. 113).

After a half rotation of the coin on the left around the central coin (of the same radius), the coin undergoes a complete rotation. In other words, a coin makes two complete rotations when rolled around the boundary of an identical coin. This fact is readily apparent in the generation of the cardioid as one disk rolling on another.

A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a diameter are called small circles. A great circle becomes a straight line in a gnomonic projection (Steinhaus 1999, pp. 220-221).The shortest path between two points on a sphere, also known as an orthodrome, is a segment of a great circle. To find the great circle (geodesic) distance between two points located at latitude and longitude of and on a sphere of radius , convert spherical coordinates to Cartesian coordinates using(1)(Note that the latitude is related to the colatitude of spherical coordinates by , so the conversion to Cartesian coordinates replaces and by and , respectively.) Now find the angle between and using the dot product,(2)(3)(4)The great circle distance is then(5)For the Earth, the equatorial radius is km, or 3963 (statute) miles. Unfortunately, the flattening..

Two points and are symmetric with respect to a circle or straight line if all circles and straight lines passing through and are orthogonal to . Möbius transformations preserve symmetry. Let a straight line be given by a point and a unit vector , thenwhere is the complex conjugate. Let a circle be given by center and radius , then

The triangulation point of a reference triangle for which triangles , , and have congruent incircles. It is a special case of an Elkies point. Kimberling and Elkies (1987) showed that a unique such point exists for any triangle, but did not provide explicit constructions for this point.

Let five circles with concyclic centers be drawn such that each intersects its neighbors in two points, with one of these intersections lying itself on the circle of centers. By joining adjacent pairs of the intersection points which do not lie on the circle of center, an (irregular) pentagram is obtained each of whose five vertices lies on one of the circles with concyclic centers.Let the circle of centers have radius and let the five circles be centered and angular positions along this circle. The radii of the circles and their angular positions along the circle of centers can then be determined by solving the ten simultaneous equations (1)(2)for , ..., 5, where and .

Four or more points , , , , ... which lie on a circle are said to be concyclic. Three points are trivially concyclic since three noncollinear points determine a circle (i.e., every triangle has a circumcircle). Ptolemy's theorem can be used to determine if four points are concyclic.The number of the lattice points which can be picked with no four concyclic is (Guy 1994).A theorem states that if any four consecutive points of a polygon are not concyclic, then its area can be increased by making them concyclic. This fact arises in some proofs that the solution to the isoperimetric problem is the circle.

Given the incircle and circumcircle of a bicentric polygon of sides, the centroid of the tangent points on the incircle is a fixed point , known as the Weill point, independent of the particular polygon.More generally, the locus of the centroid of any number of the points is a circle (Casey 1888).

The minimal enclosing circle problem, sometimes also known as the bomb problem, is the problem of finding the circle of smallest radius that contains a given set of points in its interior or on its boundary. This smallest circle is known as the minimal enclosing circle.Jung's theorem states that every finite set of points with geometric span has an enclosing circle with radius no greater than .

Concentric circles are circles with a common center. The region between two concentric circles of different radii is called an annulus. Any two circles can be made concentric by inversion by picking the inversion center as one of the limiting points.Given two concentric circles with radii and , what is the probability that a chord chosen at random from the outer circle will cut across the inner circle? Depending on how the "random" chord is chosen, 1/2, 1/3, or 1/4 could all be correct answers. 1. Picking any two points on the outer circle and connecting them gives 1/3. 2. Picking any random point on a diagonal and then picking the chordthat perpendicularly bisects it gives 1/2. 3. Picking any point on the large circle, drawing a line to the center, and thendrawing the perpendicularly bisected chord gives 1/4. So some care is obviously needed in specifying what is meant by "random" in this problem.Given an arbitrary chord to the..

Four circles may be drawn through an arbitrary point on a torus. The first two circles are obvious: one is in the plane of the torus and the second perpendicular to it. The third and fourth circles (which are inclined with respect to the torus) are much more unexpected and are known as the Villarceau circles (Villarceau 1848, Schmidt 1950, Coxeter 1969, Melzak 1983).To see that two additional circles exist, consider a coordinate system with origin at the center of torus, with pointing up. Specify the position of by its angle measured around the tube of the torus. Define for the circle of points farthest away from the center of the torus (i.e., the points with ), and draw the x-axis as the intersection of a plane through the z-axis and passing through with the -plane. Rotate about the y-axis by an angle , where(1)In terms of the old coordinates, the new coordinates are(2)(3)So in coordinates, equation (◇) of the torus becomes(4)Expanding the left..

The integral of over the unit disk is given by(1)(2)(3)(4)In general,(5)provided .Additional integrals include(6)(7)(8)

The three circumcircles through the triangle centroid of a given triangle and the pairs of the vertices of the second Brocard triangle are called the McCay circles (Johnson 1929, p. 306).The circumcircle of their centers (i.e., of the second Brocard triangle) is therefore the Brocard circle.The -McCay circle has center functionand radius,1/3 that of the Neuberg circle, where is the Brocard angle (Johnson 1929, p. 307).If the polygon vertex of a triangle describes a Neuberg circle , then its triangle centroid describes one of the McCay circles (Johnson 1929, p. 290). In the above figure, the inner triangle is the second Brocard triangle of , whose two indicated edges are concyclic with on the McCay circle.

A disk with radius 1.The (open) unit disk can also be considered to be the region in the complex plane defined by , where denotes the complex modulus. (The closed unit disk is similarly defined as .

Let , , , and be four circles of general position through a point . Let be the second intersection of the circles and . Let be the circle . Then the four circles , , , and all pass through the point . Similarly, let be a fifth circle through . Then the five points , , , and all lie on one circle . And so on.

The triquetra is a geometric figure consisting of three mutually intersecting vesica piscis lens shapes, as illustrated above. The central region common to all three lenses is a Reuleaux triangle.The triquetra has perimeterand its interior has area

While some authors define "circumference" as distance around an arbitrary closed object (sometimes restricted to a closed curved object), in the work, the term "perimeter" is used for this purpose and "circumference" is restricted to mean the perimeter of a circle.For radius or diameter ,where is pi.The term "circumference" is also sometimes used to refer to the "enclosingboundary" itself of a curved lamina or disk.

The circumcenter is the center of a triangle's circumcircle. It can be found as the intersection of the perpendicular bisectors. The trilinear coordinates of the circumcenter are(1)and the exact trilinear coordinatesare therefore(2)where is the circumradius, or equivalently(3)The circumcenter is Kimberling center .The distance between the incenter and circumcenter is , where is the circumradius and is the inradius.Distances to a number of other named triangle centers are given by(4)(5)(6)(7)(8)(9)(10)(11)(12)where is the triangle triangle centroid, is the orthocenter, is the incenter, is the symmedian point, is the nine-point center, is the Nagel point, is the de Longchamps point, is the circumradius, is Conway triangle notation, and is the triangle area.If the triangle is acute, the circumcenter is in the interior of the triangle. In a right triangle, the circumcenter is the midpoint of the hypotenuse.For an acute triangle,(13)where..

Two circles with centers at with radii for are mutually tangent if(1)If the center of the second circle is inside the first, then the and signs both correspond to internally tangent circles. If the center of the second circle is outside the first, then the sign corresponds to externally tangent circles and the sign to internally tangent circles.Finding the circles tangent to three given circles is known as Apollonius' problem. The Desborough Mirror, a beautiful bronze mirror made during the Iron Age between 50 BC and 50 AD, consists of arcs of circles that are exactly tangent (Wolfram 2002, pp. 43 and 873).Given three distinct noncollinear points , , and , denote the side lengths of the triangle as , , and . Now let three circles be drawn, one centered about each point and each one tangent to the other two (left figure), and call the radii , , .Interestingly, the pairwise external similitude centers of these circles are the three Nobbs points (P. Moses,..

The area of the dodecagon () inscribed in a unit circle with is

Select three points at random on the circumference of a unit circle and find the distribution of areas of the resulting triangles determined by these three points.The first point can be assigned coordinates without loss of generality. Call the central angles from the first point to the second and third and . The range of can be restricted to because of symmetry, but can range from . Then(1)so(2)(3)Therefore,(4)(5)(6)(7)But(8)(9)(10)(11)Write (10) as(12)then(13)and(14)From (12),(15)(16)(17)(18)(19)so(20)Also,(21)(22)(23)(24)so(25)Combining (◇) and (◇) gives the meantriangle area as(26)(OEIS A093582).The first few moments are(27)(28)(29)(30)(31)(32)(OEIS A093583 and A093584and OEIS A093585 and A093586).The variance is therefore given by(33)The probability that the interior of the triangle determined by the three points picked at random on the circumference of a circle contains the origin is 1/4...

In the figure above with tangent line and secant line ,(1)(Jurgensen et al. 1963, p. 346).The line tangent to a circle of radius centered at (2)(3)through can be found by solving the equation(4)giving(5)Two of these four solutions give tangent lines, as illustrated above, and the lengths of these lines are equal (Casey 1888, p. 29).

Let three equal circles with centers , , and intersect in a single point and intersect pairwise in the points , , and . Then the circumcircle of the reference triangle is congruent to the original three.Furthermore, the points , , , and form an orthocentric system.Here, the original three circles are known as Johnson circles and the triangle formed by their centers is known as the Johnson triangle. Amazingly, the Johnson triangle circumcircle is also congruent to the circumcircle of the reference triangle and centered at the orthocenter .A "triquetra" is a figure consisting of three circular arcs of equal radius, and has seen extensive use in heraldry (i.e., coats of arms), specifically in the case of the so-called Borromean rings. The term "Triquetra theorem" was used by Mackenzie (1992) to describe Johnson's theorem.Mackenzie (1992) generalized this theorem to the case where the three circles do not coincide. In this..

The radial curve of a unit circle from a radial point and parametric equations(1)(2)is another circle with parametricequations(3)(4)

The Johnson midpoint is the point of concurrence of the line segments joining the vertices of a reference triangle with the centers of a certain set of circles (that resemble but are not the Johnson circles). It also is the midpoint of each of these segments, as well as perspector of the reference triangle and the triangle determined by the centers of these circles.It has triangle center functionand is Kimberling center .

Johnson's theorem states that if three equal circles mutually intersect one another in a single point, then the circle passing through their other three pairwise points of intersection is congruent to the original three circles. If the pairwise intersections are taken as the vertices of a reference triangle , then the Johnson circles that are congruent to the circumcircle of have centers(1)(2)(3)where , , , and are Conway triangle notation.The centers of the Johnson circles form the Johnson triangle which, together with , form an orthocentric system.The point of threefold concurrence of the Johnson circles is the orthocenter of the reference triangle .Note also that intersections of the directed lines from the orthocenter of the reference triangle through the centers of the Johnson circles intersect the Johnson circles at the vertices of the anticomplementary triangle . The anticomplementary circle, with center and radius (where is..

A uniform distribution of points on the circumference of a circle can be obtained by picking a random real number between 0 and . Picking random points on a circle is therefore a great deal more straightforward than sphere point picking. random points can be picked on a unit circle in the Wolfram Language using the function RandomPoint[Circle[], n].Random points on a circle can also be obtained by picking two numbers , from a uniform distribution on , and rejecting pairs with . From the remaining points, the double-angle formulas then imply that the points with Cartesian coordinates(1)(2)have the desired distribution (von Neumann 1951, Cook 1957). This method can also be extended to sphere point picking (Cook 1957). The plots above show the distribution of points for 50, 100, and 500 initial points (where the counts refer to the number of points before throwing away)...

The pedal curve of a unitcircle with parametric equation(1)(2)with pedal point is(3)(4)The pedal curve with respect to the center is thecircle itself (Gray 1997, pp. 119 and 124-135).If the pedal point is taken on the circumference (and in particular at the point ), the pedal curve is the cardioid(5)(6)and otherwise is a limaçon.

The first and second isodynamic points of a triangle can be constructed by drawing the triangle's angle bisectors and exterior angle bisectors. Each pair of bisectors intersects a side of the triangle (or its extension) in two points and , for , 2, 3. The three circles having , , and as diameters are the Apollonius circles , , and . The points and in which the three Apollonius circles intersect are the first and second isodynamic points, respectively.The two isodynamic points of a reference triangle are mutually inverse with respect to the circumcircle of (Gallatly 1913, p. 103). and have triangle center functionsrespectively. The antipedal triangles of bothpoints are equilateral and have areaswhere is the Brocard angle.The isodynamic points are isogonal conjugates of the Fermat points. They lie on the Brocard axis. The distances from either isodynamic point to the polygon vertices are inversely proportional to the sides. The pedal..

The orthotomic of the unitcircle represented by(1)(2)with a source at is(3)(4)

External (or positive) and internal (or negative) similarity points of two circles with centers and and radii and are the points and on the lines such thator

Draw an initial circle, and arrange six circles tangent to it such that they touch both the original circle and their two neighbors. Then the three lines joining opposite points of tangency are concurrent in a point. The figures above show several possible configurations (Evelyn et al. 1974, pp. 31-37).Letting the radii of three of the circles approach infinity turns three of the circles into the straight sides of a triangle and the central circle into the triangle's incircle. As illustrated above, the three lines connecting opposite points of tangency (with those along the triangle edges corresponding to the vertices of the contact triangle) concur (Evelyn et al. 1974, pp. 39 and 42).

Given triangle , there are four lines simultaneously tangent to the incircle (with center ) and the -excircle (with center ). Of these, three correspond to the sidelines of the triangle, and the fourth is known as the -intangent (Kimberling 1998, p. 161), illustrated above.The intangents intersect one another pairwise, and their points of intersection formthe so-called intangents triangle.

For a unit circle with parametricequations(1)(2)the negative pedal curve with respect to the pedal point is(3)(4)Therefore if the point is inside the circle (), the negative pedal is an ellipse, if , it is a single point, if the point is outside the circle (), the negative pedal is a hyperbola.

The incenter is the center of the incircle for a polygon or insphere for a polyhedron (when they exist). The corresponding radius of the incircle or insphere is known as the inradius.The incenter can be constructed as the intersection of angle bisectors. It is also the interior point for which distances to the sides of the triangle are equal. It has trilinear coordinates 1:1:1, i.e., triangle center function(1)and homogeneous barycentric coordinates . It is Kimberling center .For a triangle with Cartesian vertices , , , the Cartesian coordinates of the incenter are given by(2)The distance between the incenter and circumcenter is , where is the circumradius and is the inradius, a result known as the Euler triangle formula.The incenter lies on the Nagel line and Soddy line, and lies on the Euler line only for an isosceles triangle. The incenter is the center of the Adams' circle, Conway circle, and incircle. It lies on the Darboux cubic, M'Cay cubic,..

Given a unit circle, pick two points at random on its circumference, forming a chord. Without loss of generality, the first point can be taken as , and the second by , with (by symmetry, the range can be limited to instead of ). The distance between the two points is then(1)The average distance is then given by(2)The probability density function is obtained from(3)The raw moments are then(4)(5)(6)giving the first few as(7)(8)(9)(10)(11)(OEIS A000984 and OEIS A093581 and A001803), where the numerators of the odd terms are 4 times OEIS A061549.The central moments are(12)(13)(14)giving the skewness and kurtosisexcess as(15)(16)Bertrand's problem asks for the probability that a chord drawn at random on a circle of radius has length .

The pedal curve of circleinvolute(1)(2)with the center as the pedal point is the Archimedes'spiral(3)(4)

Any one of the eight Apollonius circles of three given circles is tangent to a circle known as a Hart circle, as are the other three Apollonius circles having (1) like contact with two of the given circles and (2) unlike contact with the third.

The involute of the circle was first studied by Huygens when he was considering clocks without pendula for use on ships at sea. He used the circle involute in his first pendulum clock in an attempt to force the pendulum to swing in the path of a cycloid. For a circle of radius ,(1)(2)the parametric equation of the involute is given by(3)(4)The arc length, curvature,and tangential angle are(5)(6)(7)The Cesàro equation is(8)

The circle which touches the incircles , , , and of a circular triangle and its associated triangles. It is either externally tangent to and internally tangent to incircles of the associated triangles , , and (as in the above figure), or vice versa. The Hart circle has several properties which are analogous to the properties on the nine-point circle of a linear triangle. There are eight Hart circles associated with a given circular triangle.The Hart circle of any circular triangle and the Hart circles of the three associated triangles have a common tangent circle which touches the former in the opposite sense to that which it touches the latter (Lachlan 1893, p. 254). In addition, the circumcircle of any circular triangle is the Hart circle of the circular triangle formed by the circumcircles of the inverse associated triangles (Lachlan 1893, p. 254)...

The radical circle of three given circles is the circle having center at the radical center of the three circles and is orthogonal to all of them. (A circle with center at the radical center that is orthogonal to one of the original circles is always orthogonal to all three.)The following table summarizes the radical circle for some circle triplets.circlesradical circleexcirclesexcircles radical circleLucas circlesLucas circles radical circleMcCay circlesMcCay circles radical circlemixtilinear incirclesmixtilinear incircles radical circleNeuberg circlesNeuberg circles radical circlepower circlesde Longchamps circleStammler circlesStammler circles radical circletangent circlesincircle

Pick any two relatively prime integers and , then the circle of radius centered at is known as a Ford circle. No matter what and how many s and s are picked, none of the Ford circles intersect (and all are tangent to the x-axis). This can be seen by examining the squared distance between the centers of the circles with and ,(1)Let be the sum of the radii(2)then(3)But , so and the distance between circle centers is the sum of the circle radii, with equality (and therefore tangency) iff . Ford circles are related to the Farey sequence (Conway and Guy 1996).If , , and are three consecutive terms in a Farey sequence, then the circles and are tangent at(4)and the circles and intersect in(5)Moreover, lies on the circumference of the semicircle with diameter and lies on the circumference of the semicircle with diameter (Apostol 1997, p. 101)...

Let , , and be the lengths of the tangents to a circle from the vertices of a triangle with sides of lengths , , and . Then the condition that is tangent to the circumcircle of the triangle is thatThe theorem was discovered by Casey prior to Purser's independent discovery.

In Homogeneous coordinates , the equation of a circle isThe discriminant of this circle is defined asand the quadratic form is the basic invariant.

Given two circles, draw the tangents from the center of each circle to the sides of the other. Then the line segments and are of equal length.The theorem can be proved by brute force by setting up the nine equations(1)(2)(3)(4)(5)(6)(7)(8)(9)and using Gröbner basis to determine the polynomial equations satisfied by and while eliminate , , , , , , and . The resulting eighth-degree polynomials satisfied by and are identical, proving that .

The probability that random arcs of angular size cover the circumference of a circle completely (for a circle with unit circumference) iswhere is the floor function (Solomon 1978, p. 75). This was first given correctly by Stevens (1939), although partial results were obtains by Whitworth (1897), Baticle (1935), Garwood (1940), Darling (1953), and Shepp (1972).The probability that arcs leave exactly gaps is given by(Stevens 1939; Solomon 1978, pp. 75-76).

Given triangle , there are four lines simultaneously tangent to the - and -excircles (with centers and , respectively). Of these, three correspond to the sidelines of the triangle, and the fourth is known as the -extangent (Kimberling 1998, p. 162), illustrated above.The extangents intersect one another pairwise, and their points of intersection formthe so-called extangents triangle.

The radius of an excircle. Let a triangle have exradius (sometimes denoted ), opposite side of length and angle , area , and semiperimeter . Then(1)(2)(3)(Johnson 1929, p. 189), where is the circumradius. Let be the inradius, then(4)(5)(Casey 1888, p. 65) and(6)Some fascinating formulas due to Feuerbach are(7)(Johnson 1929, pp. 190-191).

Members of a coaxal system satisfyfor values of . Picking then gives the two circlesof zero radius, known as point circles. The two point circles , real or imaginary, are called the limiting points of the coaxal system.

If the tangents at and to the circumcircle of a triangle intersect in a point , then the circle with center and which passes through and is called the excosine circle, and cuts and in two points which are extremities of a diameter.

Consider a unit circle and a radiant point located at . There are four different regimes of caustics, illustrated above.For radiant point at , the catacaustic is the nephroid(1)(2)(Trott 2004, p. 17, mistakenly states that the catacaustic for parallel light falling on any concave mirror is a nephroid.)For radiant point a finite distance , the catacaustic is the curve(3)(4)which is apparently incorrectly described as a limaçonby Lawrence (1972, p. 207).For radiant point on the circumference of the circle (), the catacaustic is the cardioid(5)(6)with Cartesian equation(7)For radiant point inside the circle, the catacausticis a discontinuous two-part curve.If the radiant point is the origin, then the catacaustic degenerates to a single point at the origin since all rays reflect upon themselves back through the origin...

A circle is the set of points in a plane that are equidistant from a given point . The distance from the center is called the radius, and the point is called the center. Twice the radius is known as the diameter . The angle a circle subtends from its center is a full angle, equal to or radians.A circle has the maximum possible area for a given perimeter,and the minimum possible perimeter for a given area.The perimeter of a circle is called the circumference, and is given by(1)This can be computed using calculus using the formula for arc length in polar coordinates,(2)but since , this becomes simply(3)The circumference-to-diameter ratio for a circle is constant as the size of the circle is changed (as it must be since scaling a plane figure by a factor increases its perimeter by ), and also scales by . This ratio is denoted (pi), and has been proved transcendental.Knowing , the area of the circle can be computed either geometrically or using calculus. As the..

Given a parabola with parametricequations(1)(2)the negative pedal curve for a pedal point has equation(3)(4)Taking the pedal point at the origin gives(5)(6)which is a semicubical parabola. Similarly, taking gives(7)(8)which is a Tschirnhausen cubic.

The center of an excircle. There are three excenters for a given triangle, denoted , , . The incenter and excenters of a triangle are an orthocentric system.where is the circumcenter, are the excenters, and is the circumradius (Johnson 1929, p. 190). Denote the midpoints of the original triangle , , and . Then the lines , , and intersect in a point known as the mittenpunkt.

Given a circle expressed in trilinearcoordinates bya central circle is a circle such that is a triangle center and is a homogeneous function that is symmetric in the side lengths , , and (Kimberling 1998, p. 226).The following table summarizes the triangle centers whose trilinears correspond to a circle with (for some appropriate value of ). In the table, indicated a circle function that is known but which does not appear among the list of Kimberling centers. Note also that the circumcircle is not actually a central circle, since its trilinears 0:0:0 are not those of a triangle center.circleKimberlingKimberlingcenterAdams' circle*incenter anticomplementary circlethird power pointorthocenter Apollonius circleBevan circleincenter Bevan point Brocard circletriangle centroid midpoint of the Brocard diametercircumcircle-0circumcenter Conway circleincenter cosine circlesymmedian point de Longchamps circlethird power..

The pedal curve of an ellipse with parametric equations(1)(2)and pedal point is given by(3)(4)The pedal curve of an ellipse with pedal point at the focus is a circle (Hilbert and Cohn-Vossen 1999, pp. 25-26).For other pedal points, the pedal curves are more complicated, as illustrated above.

Given a chord of a circle, draw any other two chords and passing through its midpoint. Call the points where and meet and . Then is also the midpoint of . There are a number of proofs of this theorem, including those by W. G. Horner, Johnson (1929, p. 78), and Coxeter (1987, pp. 78 and 144). The latter concise proof employs projective geometry.The following proof is given by Coxeter and Greitzer (1967, p. 46). In the figure at right, drop perpendiculars and from and to , and and from and to . Write , , and , and then note that by similar triangles(1)(2)(3)so(4)(5)so . Q.E.D.

Orthogonal circles are orthogonal curves, i.e., they cut one another at right angles. By the Pythagorean theorem, two circles of radii and whose centers are a distance apart are orthogonal if(1)Two circles with Cartesian equations(2)(3)are orthogonal if(4)A theorem of Euclid states that, for the orthogonal circles in the above diagram,(5)(Dixon 1991, p. 65).The radical lines of three given circles concur in the radical center . If a circle with center cuts any one of the three circles orthogonally, it cuts all three orthogonally. This circle is called the orthogonal circle (or radical circle) of the system. The orthogonal circle is the locus of a point whose polars with respect to the three given circles are concurrent (Lachlan 1893, p. 237).The following table lists circles orthogonal to various named circle.circleorthogonal circle(s)Apollonius circleStevanović circleBevan circleStevanović circleBrocard..

The Borromean rings, also called the Borromean links (Livingston 1993, p. 10) are three mutually interlocked rings (left figure), named after the Italian Renaissance family who used them on their coat of arms. The configuration of rings is also known as a "Ballantine," and a brand of beer (right figure; Falstaff Brewing Corporation) has been brewed under this name. In the Borromean rings, no two rings are linked, so if any one of the rings is cut, all three rings fall apart. Any number of rings can be linked in an analogous manner (Steinhaus 1999, Wells 1991).The Borromean rings are a prime link. They have link symbol 06-0302, braid word , and are also the simplest Brunnian link.It turns out that rigid Borromean rings composed of real (finite thickness) tubes cannot be physically constructed using three circular rings of either equal or differing radii. However, they can be made from three congruent elliptical rings...

The diameter of a circle is the distance from a point on the circle to a point radians away, and is the maximum distance from one point on a circle to another. The diameter of a sphere is the maximum distance between two antipodal points on the surface of the sphere.If is the radius of a circle or sphere, then . The ratio of the circumference of a circle or great circle of a sphere to the diameter is pi,

There are four completely different definitions of the so-called Apollonius circles: 1. The set of all points whose distances from two fixed points are in a constant ratio (Durell 1928, Ogilvy 1990). 2. One of the eight circles that is simultaneously tangent to three given circles (i.e., a circle solving Apollonius' problem for three circles). 3. One of the three circles passing through a vertex and both isodynamic points and of a triangle (Kimberling 1998, p. 68). 4. The circle that touches all three excircles of atriangle and encompasses them (Kimberling 1998, p. 102). Given one side of a triangle and the ratio of the lengths of the other two sides, the locus of the third polygon vertex is the Apollonius circle (of the first type) whose center is on the extension of the given side. For a given triangle, there are three circles of Apollonius. Denote the three Apollonius circles (of the first type) of a triangle by , , and , and their centers..

Draw a circle that cuts three given circles perpendicularly. The solution is known as the radical circle of the given three circles. If it lies outside the three circles, then the circle with center and radius formed by the tangent from to one of the given circles intersects the given circles perpendicularly. Otherwise, if lies inside one of the circles, the problem is unsolvable.

If three circles , , and are taken in pairs, the external similarity points of the three pairs lie on a straight line. Similarly, the external similarity point of one pair and the two internal similarity points of the other two pairs lie upon a straight line, forming a similarity axis of the three circles.

Draw three circles in the plane, none of which lies completely inside another, and the common external tangent lines for each pair. Then points of intersection of the three pairs of tangent lines lie on a straight line.Monge's theorem has a three-dimensional analog which states that the apexes of the cones defined by four spheres, taken two at a time, lie in a plane (when the cones are drawn with the spheres on the same side of the apex; Wells 1991).

An infinite sequence of circles such that every four consecutive circles are mutually tangent, and the circles' radii ..., , ..., , , , , , , ..., , , ..., are in geometric progression with ratiowhere is the golden ratio (Gardner 1979ab). Coxeter (1968) generalized the sequence to spheres.

If a cyclic quadrilateral is inscribed in a circle of a coaxal system such that one pair of connectors touches another circle of the system at , then each pair of opposite connectors will touch a circle of the system ( at on , at on , at on , at on , and at on ), and the six points of contact , , , , , and will be collinear.The general theorem states that if , , ..., are any number of points taken in order on a circle of a given coaxal system so that , , ..., touch respectively fixed circles , , ..., of the system, then must touch a fixed circle of the system. Further, if , , ..., touch respectively any of the circles , , ..., , then must touch the remaining circle.

A stadium, also called a discorectangle, obround, or sausage body, is a geometric figure consisting of a rectangle with top and bottom lengths whose ends are capped off with semicircles of radius . The area of a stadium is therefore given by(1)and the perimeter by(2)Defining the ratio gives the alternate forms(3)and(4)The term is most commonly encountered in the context of stadiumbilliards.The stadium is implemented in the Wolfram Language as StadiumShape[lbrx1, y1, x2, y2, r].A stadium of revolution might be termed a capsule.

A unit circle is a circle of unit radius, i.e., of radius 1.The unit circle plays a significant role in a number of different areas of mathematics. For example, the functions of trigonometry are most simply defined using the unit circle. As shown in the figure above, a point on the terminal side of an angle in angle standard position measured along an arc of the unit circle has as its coordinates so that is the horizontal coordinate of and is its vertical component.As a result of this definition, the trigonometric functions are periodic with period .Another immediate result of this definition is the ability to explicitly write the coordinates of a number of points lying on the unit circle with very little computation. In the figure above, for example, points , , , and correspond to angles of , , , and radians, respectively, whereby it follows that , , , and . Similarly, this method can be used to find trigonometric values associated to integer multiples of..

Given two circles with one interior to the other, if small tangent circles can be inscribed around the region between the two circles such that the final circle is tangent to the first, the circles form a Steiner chain.The simplest way to construct a Steiner chain is to perform an inversion on a symmetrical arrangement on circles packed between a central circle of radius and an outer concentric circle of radius (Wells 1991). In this arrangement,(1)so the ratio of the radii for the small and large circles is(2)In addition, the radii of the circles in the ring are(3)and their centers are located at a distance(4)from the origin.To transform the symmetrical arrangement into a Steiner chain, take an inversion center which is a distance from the center of the symmetrical figure. Then the radii and of the outer and center circles become(5)(6)respectively. Equivalently, a Steiner chain results whenever the inversivedistance between the two original..

The midcircle of two given circles is the circle which would invert the circles into each other. Dixon (1991) gives constructions for the midcircle for four of the five possible configurations. In the case of the two intersecting circles, there are two midcircles.The midcircle is in the same coaxal system as thetwo given circles.The center of the midcircle(s) is one or both of the centers of similitude. 1. If one circle is inside the other, the unique midcircle has center at the internalsimilitude center. 2. If the circles are disjoint, the unique midcircle has center at the externalsimilitude center. 3. If the circles intersect in two points, there are two midcircles, one centered at each center of similitude. 4. If the circles intersect in a single point, the unique midcircle has center at the external similitude center (since the midcircle that would be centered at internal similitude center degenerates to a point). If the given circles intersect,..

A point about which inversion of two circles produced concentric circles. Every pair of distinct circles has two limiting points.The limiting points correspond to the point circles of a coaxal system, and the limiting points of a coaxal system are inverse points with respect to any circle of the system.To find the limiting point of two circles of radii and with centers separated by a distance , set up a coordinate system centered on the circle of radius and with the other circle centered at . Then the equation for the position of the center of the inverted circles with inversion center ,(1)becomes(2)(3)for the first and second circles, respectively. Setting gives(4)and solving using the quadratic equation gives the positions of the limiting points as(5)

Coaxal circles are circles whose centers are collinear and that share a common radical line. The collection of all coaxal circles is called a pencil of coaxal circles (Coxeter and Greitzer 1967, p. 35). It is possible to combine the two types of coaxal systems illustrated above such that the sets are orthogonal.Note that not all circles sharing the same radical line need be coaxal, since the lines of their centers need only be perpendicular to the radical line and therefore may not coincide.Members of a coaxal system satisfyfor values of . Picking then gives the two circlesof zero radius, known as point circles. The two point circles , real or imaginary, are called the limiting points.

The power of a fixed point with respect to a circle of radius and center is defined by the product(1)where and are the intersections of a line through with the circle. The term "power" was first used in this way by Jacob Steiner (Steiner 1826; Coxeter and Greitzer 1967, p. 30). Amazingly, (sometimes written ) is independent of the choice of the line (Coxeter 1969, p. 81).Now consider a point not necessarily on the circumference of the circle. If is the distance between and the circle's center , then the power of the point relative to the circle is(2)If is outside the circle, its power is positive and equal to the square of the length of the segment from to the tangent to the circle through ,(3)If lies along the x-axis, then the angle around the circle at which lies is given by solving(4)for , giving(5)for coordinates(6)The points and are inverse points, also called polar reciprocals, with respect to the inversion circle if(7)(Wenninger..

The inverse curve of the circle with parametric equations(1)(2)with respect to an inversion circle with center and radius is given by(3)(4)which is another circle.

For every positive integer , there exists a circle in the plane having exactly lattice points on its circumference. The theorem is based on the number of integral solutions to the equation(1)given by(2)where is the number of divisors of of the form and is the number of divisors of the form . It explicitly identifies such circles (the Schinzel circles) as(3)Note, however, that these solutions do not necessarily have the smallest possible radius.

A circle having a given number of lattice points on its circumference. The Schinzel circle having lattice points is given by the equation(1)Note that these solutions do not necessarily have the smallest possible radius. For example, while the Schinzel circle centered at (1/3, 0) and with radius 625/3 has nine lattice points on its circumference, so does the circle centered at (1/3, 0) with radius 65/3.A table of minimal circles to is given by Pegg (2008).

A particle is said to be undergoing uniform circular motion if its radius vector in appropriate coordinates has the form , where(1)(2)Geometrically, uniform circular motions means that moves in a circle in the -plane with some radius at constant speed. The quantity is called the angular velocity of . The speed of is(3)and the acceleration of P has constant magnitude(4)and is directed toward the center of the circle traced by . This is called centripetal acceleration.Ignoring the ellipticity of their orbits, planet show nearly uniform circular motion about the Sun. (Although due to orbital inclinations, the orbital planes of the different planets are not necessarily coplanar.)

The osculating circle of a curve at a given point is the circle that has the same tangent as at point as well as the same curvature. Just as the tangent line is the line best approximating a curve at a point , the osculating circle is the best circle that approximates the curve at (Gray 1997, p. 111).Ignoring degenerate curves such as straight lines, the osculating circle of a given curve at a given point is unique.Given a plane curve with parametric equations and parameterized by a variable , the radius of the osculating circle is simply the radius of curvature(1)where is the curvature, and the center is just the point on the evolute corresponding to ,(2)(3)Here, derivatives are taken with respect to the parameter .In addition, let denote the circle passing through three points on a curve with . Then the osculating circle is given by(4)(Gray 1997)...

The goat problem (or bull-tethering problem) considers a fenced circular field of radius with a goat (or bull, or other animal) tied to a point on the interior or exterior of the fence by means of a tether of length , and asks for the solution to various problems concerning how much of the field can be grazed.Tieing a goat to a point on the interior of the fence with radius 1 using a chain of length , consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection(1)Taking gives(2)plotted above. Setting (i.e., half of ) leads to the equation(3)which cannot be solved exactly, but which has approximate solution(4)(OEIS A133731).Now instead consider tieing the goat to the exterior of the fence (or equivalently, to the exterior of a silo whose horizontal cross section is a circle) with radius . Assume that , so that the goat is not..

The region lying between two concentric circles. The area of the annulus formed by two circles of radii and (with ) isThe annulus is implemented in the Wolfram Language as Annulus[x, y, b, a].In the above figure, the area of the circle whose diameter is tangent to the inner circle and has endpoints at the outer circle is equal to the area of the annulus.

Let a circle lie inside another circle . From any point on , draw a tangent to and extend it to . From the point, draw another tangent, etc. For tangents, the result is called an -sided Poncelet transverse.If, on the circle of circumscription there is one point of origin for which a four-sided Poncelet transverse is closed, then the four-sided transverse will also close for any other point of origin on the circle (Dörrie 1965).

If an -sided Poncelet transverse constructed for two given conic sections is closed for one point of origin, it is closed for any position of the point of origin. Specifically, given one ellipse inside another, if there exists one circuminscribed (simultaneously inscribed in the outer and circumscribed on the inner) -gon, then any point on the boundary of the outer ellipse is the vertex of some circuminscribed -gon. If the conic is taken as a circle (Casey 1888, pp. 124-126) , then a polygon which has both an incenter and a circumcenter (and for which the transversals would therefore close) is called a bicentric polygon.Amazingly, this problem is isomorphic to Gelfand'squestion (King 1994).For an even-sided polygon, the diagonals are concurrent at the limiting point of the two circles, whereas for an odd-sided polygon, the lines connecting the vertices to the opposite points of tangency are concurrent at the limiting point.Inverting..

Sangaku problems, often written "san gaku," are geometric problems of the type found on devotional mathematical wooden tablets ("sangaku") which were hung under the roofs of shrines or temples in Japan during two centuries of schism from the West (Fukagawa and Pedoe 1989). During the time of isolation, Japanese mathematicians developed their own "traditional mathematics," which, in the 1850s, began giving way to Western methods. There were also changes in the script in which mathematics was written and, as a result, few people now living know how to interpret the historic tablets (Kimberling).Japanese mathematicians represented in sangaku include Seki Kowa (1642-1708), Ajima Chokuen (also called Naonobu; 1732-1798), and Shoto Kenmotu (1790-1871).Sangaku problems typically involve mutually tangent circles or tangent spheres, with specific examples including the properties of the Ajima-Malfatti..

Starting with the circle tangent to the three semicircles forming the arbelos, construct a chain of tangent circles , all tangent to one of the two small interior circles and to the large exterior one. This chain is called the Pappus chain (left figure).In a Pappus chain, the distance from the center of the first inscribed circle to the bottom line is twice the circle's radius, from the second circle is four times the radius, and for the th circle is times the radius. Furthermore, the centers of the circles lie on an ellipse (right figure).If , then the center and radius of the th circle in the Pappus chain are(1)(2)(3)This general result simplifies to for (Gardner 1979). Further special cases when are considered by Gaba (1940).The positions of the points of tangency for the first circle are(4)(5)(6)(7)(8)(9)The diameter of the th circle is given by ()th the perpendicular distance to the base of the semicircle. This result was known to Pappus, who referred..

Let a convex cyclic polygon be triangulated in any manner, and draw the incircle to each triangle so constructed. Then the sum of the inradii is a constant independent of the triangulation chosen. This theorem can be proved using Carnot's theorem. In the above figures, for example, the inradii of the left triangulation are 0.142479, 0.156972, 0.232307, 0.498525, and the inradii of the right triangulation are 0.157243, 0.206644, 0.312037, 0.354359, giving a sum of 1.03028 in each case.According to an ancient custom of Japanese mathematicians, this theorem was a Sangaku problem inscribed on tablets hung in a Japanese temple to honor the gods and the author in 1800 (Johnson 1929).The converse is also true: if the sum of inradii does not depend on the triangulation of a polygon, then the polygon is cyclic...

A special case of Apollonius' problem requiring the determination of a circle touching three mutually tangent circles (also called the kissing circles problem). There are two solutions: a small circle surrounded by the three original circles, and a large circle surrounding the original three. Frederick Soddy gave the formula for finding the radius of the so-called inner and outer Soddy circles given the radii of the other three. The relationship iswhere are the curvatures of the circles with radii . Here, the negative solution corresponds to the outer Soddy circle and the positive solution to the inner Soddy circle.This formula was known to Descartes and Viète (Boyer and Merzbach 1991, p. 159), but Soddy extended it to spheres. In -dimensional space, mutually touching -spheres can always be found, and the relationship of their curvatures is..

Four circles , , , and are tangent to a fifth circle or a straight line iff(1)where is the length of a common tangent to circles and (Johnson 1929, pp. 121-122). The following cases are possible: 1. If all the s are direct common tangents, then has like contact with all the circles, 2. If the s from one circle are transverse while the other three are direct, then this one circle has contact with unlike that of the other three, 3. If the given circles can be so paired that the common tangents to the circles of each pair are direct, while the other four are transverse, then the members of each pair have like contact with (Johnson 1929, p. 125).The special case of Casey's theorem shown above was given in a Sangaku problem from 1874 in the Gumma Prefecture. In this form, a single circle is drawn inside a square, and four circles are then drawn around it, each of which is tangent to the square on two of its sides. For a square of side length with lower left corner..

Jung's theorem states that the generalized diameter of a compact set in satisfieswhere is the circumradius of (Danzer et al. 1963).This gives the special case that every finite set of points in two dimensions with geometric span has an enclosing circle with radius no greater than (Rademacher and Toeplitz 1957, p. 104).

In 1803, Malfatti posed the problem of determining the three circular columns of marble of possibly different sizes which, when carved out of a right triangular prism, would have the largest possible total cross section. This is equivalent to finding the maximum total area of three circles which can be packed inside a right triangle of any shape without overlapping. This problem is now known as the marble problem (Martin 1998, p. 92). Malfatti gave the solution as three circles (the Malfatti circles) tangent to each other and to two sides of the triangle. In 1930, it was shown that the Malfatti circles were not always the best solution. Then Goldberg (1967) showed that, even worse, they are never the best solution (Ogilvy 1990, pp. 145-147). Ogilvy (1990, pp. 146-147) and Wells (1991) illustrate specific cases where alternative solutions are clearly optimal.The general Malfatti problem on an arbitrary triangle was actually..

Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles. The Malfatti configuration appears on the cover of Martin (1998).The positions and radii of the Malfatti circles can be found by labeling sides and distances as illustrated above. The length of the projection of the line connecting circles and onto side can be found from the diagram at right to be(1)(2)Therefore, three equations follow from the condition that the labeled lengths must sum to the side lengths,(3)(4)(5)Three additional equations follow from the fact that the circle centers lie on the corresponding angle bisectors of the triangle vertices, so(6)(7)(8)Re-expressing these equations in terms of side lengths and rearranging and squaring to eliminate square roots then gives the system of six polynomial equations (9)(10)(11)(12)(13)(14)This system can be solved simultaneously for the..

A circle packing is called rigid (or "stable") if every circle is fixed by its neighbors, i.e., no circle can be translated without disturbing other circles of the packing (e.g., Niggli 1927, Niggli 1928, Fejes Tóth 1960/61). Böröczky (1964) exhibited stable systems of congruent unit circles with density 0. A rigid packing of circles can be obtained from a hexagonal tessellation by removing the centers of a hexagonal web, then replacing each remaining circle with three equal inscribed circles (appropriately oriented), as illustrated above (Meschkowski 1966, Wells 1991).

A circle packing is an arrangement of circles inside a given boundary such that no two overlap and some (or all) of them are mutually tangent. The generalization to spheres is called a sphere packing. Tessellations of regular polygons correspond to particular circle packings (Williams 1979, pp. 35-41). There is a well-developed theory of circle packing in the context of discrete conformal mapping (Stephenson).The densest packing of circles in the plane is the hexagonal lattice of the bee's honeycomb (right figure; Steinhaus 1999, p. 202), which has a packing density of(1)(OEIS A093766; Wells 1986, p. 30). Gauss proved that the hexagonal lattice is the densest plane lattice packing, and in 1940, L. Fejes Tóth proved that the hexagonal lattice is indeed the densest of all possible plane packings.Surprisingly, the circular disk is not the least economical region for packing the plane. The "worst"..

Consider three mutually tangent circles, and draw their inner Soddy circle. Then draw the inner Soddy circles of this circle with each pair of the original three, and continue iteratively. The steps in the process are illustrated above (Trott 2004, pp. 34-35).An animation illustrating the construction of the gasket is shown above.The points which are never inside a circle form a set of measure 0 having fractal dimension approximately 1.3058 (Mandelbrot 1983, p. 172). The Apollonian gasket corresponds to a limit set that is invariant under a Kleinian group (Wolfram 2002, p. 986).The Apollonian gasket can also be generalized to three dimensions (Boyd 1973, Andrade et al. 2005), as illustrated above. A graph obtained by connecting the centers of touching spheres in a three-dimensional Apollonian gasket by edges is known as an Apollonian network...

A problem sometimes known as Moser's circle problem asks to determine the number of pieces into which a circle is divided if points on its circumference are joined by chords with no three internally concurrent. The answer is(1)(2)(Yaglom and Yaglom 1987, Guy 1988, Conway and Guy 1996, Noy 1996), where is a binomial coefficient. The first few values are 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... (OEIS A000127). This sequence demonstrates the danger in making assumptions based on limited trials. While the series starts off like , it begins differing from this geometric series at .

Determining the maximum number of pieces in which it is possible to divide a circle for a given number of cuts is called the circle cutting or pancake cutting problem. The minimum number is always , where is the number of cuts, and it is always possible to obtain any number of pieces between the minimum and maximum. The first cut creates 2 regions, and the th cut creates new regions, so(1)(2)(3)Therefore,(4)(5)(6)(7)(8)Evaluating for , 2, ... gives 2, 4, 7, 11, 16, 22, ... (OEIS A000124). This is equivalent to the maximal number of regions into which a plane can be cut by lines.

If a circular pizza is divided into 8, 12, 16, ... slices by making cuts at equal angles from an arbitrary point, then the sums of the areas of alternate slices are equal.There is also a second pizza theorem. This one gives the volume of a pizza of thickness and radius :

Count the number of lattice points inside the boundary of a circle of radius with center at the origin. The exact solution is given by the sum(1)(2)(3)(Hilbert and Cohn-Vossen 1999, p. 39). The first few values for , 1, ... are 1, 5, 13, 29, 49, 81, 113, 149, ... (OEIS A000328).The series for is intimately connected with the sum of squares function (i.e., the number of representations of by two squares), since(4)(Hardy 1999, p. 67). is also closely connected with the Leibniz series since(5)where is a Lerch transcendent and is a digamma function, so taking the limit gives(6)(Hilbert and Cohn-Vossen 1999, p. 39).Gauss showed that(7)where(8)(Hardy 1999, p. 67).The first few values of are 5, 13/4, 29/9, 49/16, 81/25, 113/36, 149/49, 197/64, 253/81, 317/100, 377/121, 49/16, ... (OEIS A000328 and A093837). As can be seen in the plot above, the values of such that are , 3, 4, 6, 11, 16, 21, 36, 52, 53, 86, 101, ... (OEIS A093832).Writing..

There are two incompatible definitions of the squircle.The first defines the squircle as the quartic plane curve which is special case of the superellipse with and , namely(1)illustrated above. This curve as arc length(2)(3)(OEIS A186642), where is a Meijer G-function (M. Trott, pers. comm., Oct. 21, 2011), encloses area(4)and has area moment of inertia tensor(5)The second definition of the squircle was given by Fernandez Guasti (1992), but apparently not dubbed with the name "squircle" until later (Fernández Guasti et al. 2005). This curve has quartic Cartesian equation(6)with squareness parameter , where corresponds to a circle with radius and to a square of side length . This curve is actually semialgebraic, as it must be restricted to to exclude other branches. This squircle encloses area(7)where is an elliptic integral of the second kind, which can be verified reduces to for and for .Both versions..

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