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Angle bisector

The (interior) bisector of an angle, also called the internal angle bisector (Kimberling 1998, pp. 11-12), is the line or line segment that divides the angle into two equal parts.The angle bisectors meet at the incenter , which has trilinear coordinates 1:1:1.The length of the bisector of angle in the above triangle is given bywhere and .The points , , and have trilinear coordinates , , and , respectively, and form the vertices of the incentral triangle.

Pascal lines

The lines containing the three points of the intersection of the three pairs of oppositesides of a (not necessarily regular) hexagon.There are 6! (i.e., 6 factorial) possible ways of taking all polygon vertices in any order, but among these are six equivalent cyclic permutations and two possible orderings, so the total number of different hexagons (not all simple) isThere are therefore a total of 60 Pascal lines created by connecting polygonvertices in any order.The 60 Pascal lines form a very complicated pattern which can be visualized most easily in the degenerate case of a regular hexagon inscribed in a circle, as illustrated above for magnifications ranging over five powers of 2. Only 45 lines are visible in this figure since each of the three thick lines (located at angles to each other) represents a degenerate group of four Pascal lines, and six of the Pascal lines are lines at infinity (Wells 1991). The pattern for a general ellipse and hexagon..

Diagonal paradox

Consider the length of the diagonal of a unit square as approximated by piecewise linear steps that may only be taken in the right and up directions. Obviously, the length so obtained is equal to half the perimeter, or 2. As the number of steps becomes large, the path visually appears to approach a diagonal line. However, no matter how small the steps, if they are constrained to be only to the right and up, their total length is always 2, despite the fact that the length of the diagonal is .This apparent paradox arises in physics in the computation of Feynman diagrams, where it has implications for the types of paths that must be included in order to obtain a good approximation to physical quantities.

Perpendicular foot

The perpendicular foot, also called the foot of an altitude, is the point on the leg opposite a given vertex of a triangle at which the perpendicular passing through that vertex intersects the side. The length of the line segment from the vertex to the perpendicular foot is called the altitude of the triangle.When a line is drawn from a point to a plane,its intersection with the plane is known as the foot.


Whirls are figures constructed by nesting a sequence of polygons (each having the same number of sides), each slightly smaller and rotated relative to the previous one. The vertices give the path of the mice in the mice problem, and form logarithmic spirals.The square whirl appears on the cover of Freund (1993).

Vertex figure

The vertex figure at a vertex of a polygon is the line segment joining the midpoints of the two adjacent sides meeting at . For a regular -gon with side length , the length of the vertex figure isThe vertex figure at a vertex of a polyhedron is the polygon whose sides are the vertex figures of the faces surrounding . The faces that join at a polyhedron vertex form a solid angle whose section by the plane is the vertex figure, as illustrated above for one vertex of the cube.The vertex figures of the Platonic solids yield the polyhedra (with holes centered on the centroids of the original faces) have convex hulls illustrated above and summarized in the following table.polyhedronconvex hull of vertex figurescubecuboctahedrondodecahedronicosidodecahedronicosahedronicosidodecahedronoctahedroncuboctahedrontetrahedronoctahedronThe illustrations above show the Archimedean solids, their vertex figures, and the solids obtained by taking..

Pierpont prime

A Pierpont prime is a prime number of the form . The first few Pierpont primes are 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, ... (OEIS A005109).A regular polygon of sides can be constructed by ruler, compass and angle-trisector iffwhere , , ..., are distinct Pierpont primes and (Gleason 1998).The numbers of Pierpont primes less than , , ... are 4, 10, 18, 25, 32, 42, 50, 58, ... (OEIS A113420) and the number less than , , , , ... are 4, 10, 25, 58, 125, 250, 505, 1020, 2075, 4227, ... (OEIS A113412; Caldwell).As of Apr. 2010, the largest known Pierpont prime is , which has decimal digits (https://primes.utm.edu/primes/page.php?id=87449).

Nested polygon

Beautiful patterns can be created by drawing sets of nested polygons such that the incircle of the th polygon is the circumcircle of the st and successive polygons are rotated one half-turn at each iteration. The results are shown above for nested triangles through heptagons in alternating black and white and in a cyclic rainbow coloring.The animation above shows successive iterations of a nested octagon.The black region of a nested square has areaif the initial square has unit edge length.

Constructible polygon

Compass and straightedge geometric constructions dating back to Euclid were capable of inscribing regular polygons of 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, ..., sides. In 1796 (when he was 19 years old), Gauss gave a sufficient condition for a regular -gon to be constructible, which he also conjectured (but did not prove) to be necessary, thus showing that regular -gons were constructible for , 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, ... (OEIS A003401).A complete enumeration of "constructible" polygons is given by those with central angles corresponding to so-called trigonometry angles.Gardner (1977) and independently Watkins (Conway and Guy 1996, Krížek et al. 2001) noticed that the number of sides for constructible polygons with odd numbers of sides are given by the first 32 rows of the Sierpiński sieve interpreted as binary numbers, giving 1, 3, 5, 15, 17, 51, 85,..

Bill picture

A Bill picture is a sequence of nested regular polygons in which subsequent polygons are each rotated so that they begin one vertex further. The term was coined by Trott (2004, pp. 88-89) and commemorates Swiss artist Max Bill, who in 1938 created a picture showing a similar arrangement of the equilateral triangle through octagon (Huttingerr 1978, Bill 1987).The figure above shows the Bill picture including regular polygons up through theregular dodecagon.

Polyhex tiling

There are no tilings of the equilateral triangle of side length 7 by all the polyhexes of order . There are nine distinct solutions of all the polyhexes of order which tile a parallelogram of base length 7 and side length 4, one of which is illustrated above (Beeler 1972).

Truchet tiling

In 1704, Sebastien Truchet considered all possible patterns formed by tilings of right triangles oriented at the four corners of a square (Wolfram 2002, p. 875).Truchet's tiles produce beautiful patterns when laid out on a grid, as illustrated by the arrangement of random tiles illustrated above.A modification of Truchet's tiles leads to a single tile consisting of two circular arcs of radius equal to half the tile edge length centered at opposed corners (Pickover 1989). There are two possible orientations of this tile, and tiling the plane using tiles with random orientations gives visually interesting patterns. In fact, these tiles have been used in the construction of various games, including the "black path game" and "meander" (Berlekamp et al. 1982, pp. 682-684).The illustration above shows a Truchet tiling. For random orientations, the fraction of closed circles is approximately 0.054 and the..

Great circle

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..

Newton's formulas

Let a triangle have side lengths , , and with opposite angles , , and . Then(1)(2)(3)


The line joining the three collinear points of intersection of the extensions of corresponding sides in perspective triangles, also called the perspective axis or homology axis.The following table summarizes the perspectrices for various pairs of named triangles. Pairs in which one triangle is inscribed in another have the line at infinity as their perspectrix and are omitted in the list below.triangle 1triangle 2perspectrixKimberlinglineanticomplementary trianglecircum-medial trianglethird power pointde Longchamps lineanticomplementary trianglecircum-orthic triangle***anticomplementary trianglecontact triangle***anticomplementary triangleD-triangle***anticomplementary triangleextouch triangle***anticomplementary trianglefirst Brocard triangle***anticomplementary trianglefirst Neuberg triangle***anticomplementary trianglefirst Yff triangle***anticomplementary triangleFuhrmann..

Perspective triangles

Two triangles and are said to be perspective, or sometimes homologic, from a line if the extensions of their three pairs of corresponding sides meet in collinear points , , and . The line joining these points is called the perspectrix.Two triangles are perspective from a point if their three pairs of corresponding polygon vertices are joined by lines which meet in a point of concurrence . This point is called the perspector, perspective center, homology center, or pole.Desargues' theorem guarantees that if two triangles are perspective from a point, they are perspective from a line (called the perspectrix). Triangles in perspective are sometimes said to be homologous or copolar.

Sondat's theorem

The perspectrix of a pair of paralogic triangles and bisects the line joining the two orthocenters and (Johnson 1929, p. 259).

Triangle point picking

Given a triangle with one vertex at the origin and the others at positions and , one might think that a random point inside the triangle would be given by(1)where and are uniform variates in the interval . However, as can be seen in the plot above, this samples the triangle nonuniformly, concentrating points in the corner.Randomly picking each of the trilinear coordinates from a uniform distribution also does not produce a uniform point spacing on in the triangle. As illustrated above, the resulting points are concentrated towards the center.To pick points uniformly distributed inside the triangle, instead pick(2)where and are uniform variates in the interval , which gives points uniformly distributed in a quadrilateral (left figure). The points not in the triangle interior can then either be discarded, or transformed into the corresponding point inside the triangle (right figure).The expected distance of a point picked at random inside..

Triangle line picking

Consider the average length of a line segment determined by two points picked at random in the interior of an arbitrary triangle. This problem is not affine, so a simple formula in terms of the area or linear properties of the original triangle apparently does not exist.However, if the original triangle is chosen to be an isosceles right triangle with unit legs, then the average length of a line with endpoints chosen at random inside it is given by(1)(2)(3)(OEIS A093063; M. Trott, pers. comm., Mar. 10, 2004), which is numerically surprisingly close to .Similarly, if the original triangle is chosen to be an equilateral triangle with unit side lengths, then the average length of a line with endpoints chosen at random inside it is given by(4)(5)The integrand can be split up into the four pieces(6)(7)(8)(9)As illustrated above, symmetry immediately gives and , so(10)With some effort, the integrals and can be done analytically to give..

Triangle circumscribing

Every convex body in the Euclidean plane with area can be inscribed in a triangle of area at most equal to (Gross 1918, Eggleston 1957). The worst possible fit corresponds (exclusively) to the case that is a parallelogram.

Domino tiling

The Fibonacci number gives the number of ways for dominoes to cover a checkerboard, as illustrated in the diagrams above (Dickau).The numbers of domino tilings, also known as dimer coverings, of a square for , 2, ... are given by 2, 36, 6728, 12988816, ... (OEIS A004003). The 36 tilings on the square are illustrated above. A formula for these numbers is given by(1)Writing(2)gives the surprising result(3)(John and Sachs 2000). For , 2, ..., the first few terms are 1, 3, 29, 5, 5, 7, 25, 9, 9, 11, 21, ... (OEIS A143234).Writing(4)(5)(6)(7)(OEIS A143233), where is Catalan's constant.

Polyomino tiling

A polyomino tiling is a tiling of the plane by specified types of polyominoes. Tiling by polyominoes has been investigated since at least the late 1950s, particularly by S. Golomb (Wolfram 2002, p. 943).Interestingly, the Fibonacci number gives the number of ways for dominoes to cover a checkerboard.Each monomino, domino, triomino, tetromino, pentomino, and hexomino tiles the plane without requiring flipping. In addition, each heptomino with the exception of the four illustrated above can tile the plane, also without flipping (Schroeppel 1972).Recently, sets of polyominoes that force non-periodic patterns have been found. The set illustrated at left above was announced by Roger Penrose in 1994, and the slightly smaller set illustrated at right below was found by Matthew Cook (Wolfram 2002, p. 943).Both of these sets yield nested patterns, as illustrated above for Cook's tiles (Wolfram2002, p. 943).Consider..

Unit square integral

Integrals over the unit square arising in geometricprobability are(1)which give the average distances in square point picking from a point picked at random in a unit square to a corner and to the center, respectively.Unit square integrals involving the absolute valueare given by(2)(3)for and , respectively.Another simple integral is given by(4)(Bailey et al. 2007, p. 67). Squaring the denominator gives(5)(6)(7)(8)(9)(OEIS A093754; M. Trott, pers. comm.), where is Catalan's constant and is a generalized hypergeometric function. A related integral is given by(10)which diverges in the Riemannian sense, as can quickly seen by transforming to polar coordinates. However, taking instead Hadamard integral to discard the divergent portion inside the unit circle gives(11)(12)(13)(14)(OEIS A093753), where is Catalan's constant.A collection of beautiful integrals over the unit squareare given by Guillera and Sondow..

Symmetric points

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

Golden gnomon

The golden gnomon is the obtuse isosceles triangle whose ratio of side to base lengths is given by , where is the golden ratio. Such a triangle has angles of -- and can be constructed from a regular pentagon as illustrated above in red. The corresponding 36-72-72 triangle with side-to-base ratio is a golden triangle.Golden triangles and gnomons can be dissected into smaller triangles that are golden gnomons and golden triangles (Livio 2003, p. 79).

Golden triangle

The golden triangle, sometimes also called the sublime triangle, is an isosceles triangle such that the ratio of the hypotenuse to base is equal to the golden ratio, . From the above figure, this means that the triangle has vertex angle equal to(1)or , and that the height is related to the base through(2)(3)(4)The inradius of a golden triangle is(5)The triangles at the tips of a pentagram (left figure) and obtained by dividing a decagon by connecting opposite vertices (right figure) are golden triangles. This follows from the fact that(6)for a pentagram and that the circumradius of a decagon of side length is(7)Golden triangles and gnomons can be dissected into smaller triangles that are golden gnomons and golden triangles (Livio 2002, p. 79).Successive points dividing a golden triangle into golden gnomons and triangles lieon a logarithmic spiral (Livio 2002, p. 119).Kimberling (1991) defines a second type of golden triangle..

Line at infinity

The straight line on which all points at infinity lie. The line at infinity is central line (Kimberling 1998, p. 150), and has trilinear equationwhich follows from the fact that a real triangle will have positive area, and therefore thatThe line at infinity passes through Kimberling centers for (the Euler infinity point),511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 674, 680, 688, 690, 696, 698, 700, 702, 704, 706, 708, 710, 712, 714, 716, 718, 720, 722, 724, 726, 730, 732, 734, 736, 740, 742, 744, 746, 752, 754, 758, 760, 766, 768, 772, 776, 778, 780, 782, 784, 786, 788, 790, 792, 794, 796, 802, 804, 806, 808, 812, 814, 816, 818, 824, 826, 830, 832, 834, 838, 888, 891, 900, 912, 916, 918, 924, 926, 928, 952, 971, 1154, 1499, 1503, 1510, 1912, 1938, 1946, 2385, 2386, 2387, 2388, 2389, 2390, 2391, 2392, 2393, 2574, 2575, 2771, 2772,..

Trilinear line

A line can be specified in trilinear coordinates by parameters such that the trilinear coordinates obey(1)The trilinear line at infinity of a triangle with side lengths , , and is(2)The line passing through points and is given by(3)(4)(5)Three trilinear points , , and are collinear if(6)Three lines(7)(8)(9)concur iff(10)in which case the point of concurrence is(11)

Cube square inscribing

What is the area of the largest square that can be inscribed on a unit cube (Trott 2004, p. 104)? The answer is 9/8, given by a square with vertices (1/4, 0, 0), (0, 1, 1/4), (3/4, 1, 1), (1, 0, 3/4), or any configuration equivalent by symmetry.In general, let be the edge of the largest -dimensional cube that fits inside an -dimensional cube, with . Then(1)(2)(3)(4)(Croft et al. 1991, p. 53). For larger , little is known.

Unit disk integral

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

Tangent circles

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,..

Circle triangle picking

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...

Circle point picking

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)...

Circle orthotomic

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

Circle line picking

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 .

Circle involute

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)

Circle covering by arcs

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).

Ellipse pedal curve

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.

Borromean rings

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...

Golden spiral

Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.In the Season 4 episode "Masterpiece" (2008) of the CBS-TV crime drama "Criminal Minds," the agents of the FBI Behavioral Analysis Unit are confronted by a serial killer who uses the Fibonacci number sequence to determine the number of victims for each of his killing episodes. In this episode, character Dr. Reid also notices that locations of the killings lie on the graph of a golden spiral, and going to the center of the spiral allows Reid to determine the location of the killer's base of operations.

Golden rectangle

Given a rectangle having sides in the ratio , the golden ratio is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio . Such a rectangle is called a golden rectangle. Euclid used the following construction to construct them. Draw the square , call the midpoint of , so that . Now draw the segment , which has length(1)and construct with this length. Now complete the rectangle , which is golden since(2)Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.The spiral is not actually tangent at these points, however, but passes through them and intersects the adjacent side, as illustrated above.If the top left corner of the original square is positioned at (0, 0), the center of the spiral occurs at the position(3)(4)(5)(6)(7)(8)(9)(10)(11)and..

Golden rhombus

A golden rhombus is a rhombus whose diagonals are in the ratio , where is the golden ratio.The faces of the acute golden rhombohedron, Bilinski dodecahedron, obtuse golden rhombohedron, rhombic hexecontahedron, and rhombic triacontahedron are golden rhombi.The half-angle is given by(1)(2)(3)(4)(OEIS A195693).Labeling the smaller interior angle as and the larger as , then(5)and(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(OEIS A105199 and A137218).The diagonal lengths of a golden rhombus with edge length are given by(18)(19)(20)(21)(22)(23)(24)(25)(OEIS A121570 and A179290),the inradius by(26)and the area by(27)

Medial parallelogram

When a pair of non-incident edges of a tetrahedron is chosen, the midpoints of the remaining 4 edges are the vertices of a planar parallelogram. Furthermore, the area of this parallelogram determined by the edges of lengths and in the figure above is given by(Yetter 1998; Trott 2004, pp. 65-66)

Triangle counting

Given rods of length 1, 2, ..., , how many distinct triangles can be made? Lengths for which(1)obviously do not give triangles, but all other combinations of three rods do. The answer is(2)The values for , 2, ... are 0, 0, 0, 1, 3, 7, 13, 22, 34, 50, ... (OEIS A002623). Somewhat surprisingly, this sequence is also given by the generating function(3)

Pythagorean fraction

Given a Pythagorean triple , the fractions and are called Pythagorean fractions. Diophantus showed that the Pythagorean fractions consist precisely of fractions of the form .

Steiner chain

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..

Stammler circles radical circle

The radical circle of the Stammler circles has center at the nine-point center , which is Kimberling center . The radius is given by(1)(2)(3)(P. Moses and J.-P. Ehrmann, pers. comm., Jan. 28, 2004), where is the circumradius, is the circumcenter, is the orthocenter, is the nine-point center, and is the area of the reference triangle.Its circle function is given by(4)which corresponds to Kimberling center .No Kimberling centers lie on it.The radical line of the circumcircle and Stammler circles radical circle passes through the circumcenter (i.e., bisects the circumcircle) and is perpendicular to the Euler line (P. Moses, pers. comm., Jan. 28, 2005).The radical line of the Stammler circle and Stammler circles radical circle passes through the Kimberling center and is perpendicular to the Euler line (P. Moses, pers. comm., Jan. 28, 2005).The radical line of the first Droz-Farny circle and..

Circle inverse curve

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.

Uniform circular motion

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.)

Brocard points

The first Brocard point is the interior point (also denoted or ) of a triangle with points labeled in counterclockwise order for which the angles , , and are equal, with the unique such angle denoted . It is not a triangle center, but has trilinear coordinates(1)(Kimberling 1998, p. 47).Note that extreme care is needed when consulting the literature, since reversing the order in which the points of the triangle are labeled results in exchanging the Brocard points.The second Brocard point is the interior point (also denoted or ) for which the angles , , and are equal, with the unique such angle denoted . It is not a triangle center, but has trilinear coordinates(2)(Kimberling 1998, p. 47).Moreover, the two angles are equal, and this angle is called the Brocard angle,(3)(4)The first two Brocard points are isogonal conjugates (Johnson 1929, p. 266). They were described by French army officer Henri Brocard in 1875, although they..

Happy end problem

The happy end problem, also called the "happy ending problem," is the problem of determining for the smallest number of points in general position in the plane (i.e., no three of which are collinear), such that every possible arrangement of points will always contain at least one set of points that are the vertices of a convex polygon of sides. The problem was so-named by Erdős when two investigators who first worked on the problem, Ester Klein and George Szekeres, became engaged and subsequently married (Hoffman 1998, p. 76).Since three noncollinear points always determine a triangle, .Random arrangements of points are illustrated above. Note that no convex quadrilaterals are possible for the arrangements shown in the fifth and eighth figures above, so must be greater than 4. E. Klein proved that by showing that any arrangement of five points must fall into one of the three cases (left top figure; Hoffman 1998,..


A homeomorphism, also called a continuous transformation, is an equivalence relation and one-to-one correspondence between points in two geometric figures or topological spaces that is continuous in both directions. A homeomorphism which also preserves distances is called an isometry. Affine transformations are another type of common geometric homeomorphism.The similarity in meaning and form of the words "homomorphism"and "homeomorphism" is unfortunate and a common source of confusion.

Home plate

Home plate in the game of baseball is an irregular pentagon with two parallel sides, each perpendicular to a base. It seems reasonable to dub such a figure (i.e., a rectangle with a coincident isosceles triangle placed on one side) a "isosceles right pentagon."However, specification of the shape of home plate, illustrated above, as specified by both the Major League Baseball Official Rules and the Little League rulebook (Kreutzer and Kerley 1990) is not physically realizable, since it requires the existence of a (12, 12, 17) right triangle, whereas(Bradley 1996). More specifically, the standards require the existence of an isosceles right triangle with side lengths 8.5 inches and a hypotenuse of length 12 inches, which does not satisfy the Pythagorean theorem.

Goat problem

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..

Railroad track problem

Given a straight segment of track of length , add a small segment so that the track bows into a circular arc. Find the maximum displacement of the bowed track. The Pythagorean theorem gives(1)But is simply , so(2)Solving (1) and (2) for gives(3)Expressing the length of the arc in terms of the centralangle,(4)(5)(6)(7)But is given by(8)so plugging in gives(9)(10)This is a transcendental equation that cannot be solved exactly with a closed-form solution for , but for ,(11)Therefore,(12)(13)Keeping only terms to order ,(14)(15)so(16)and(17)If we take and 1 foot, then feet. Solving equation (◇) numerically, we find that the true answer is feet.

Pythagoras's theorem

Pythagoras's theorem states that the diagonal of a square with sides of integral length cannot be rational. Assume is rational and equal to where and are integers with no common factors. Thensoand , so is even. But if is even, then is even. Since is defined to be expressed in lowest terms, must be odd; otherwise and would have the common factor 2. Since is even, we can let , then . Therefore, , and , so must be even. But cannot be both even and odd, so there are no and such that is rational, and must be irrational.In particular, Pythagoras's constant is irrational. Conway and Guy (1996) give a proof of this fact using paper folding, as well as similar proofs for (the golden ratio) and using a pentagon and hexagon. A collection of 17 computer proofs of the irrationality of is given by Wiedijk (2006)...

Poncelet transverse

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).

Poncelet's porism

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..

Ono inequality

Ono (1914) conjectured that the inequalityholds true for all triangles, where , , and are the lengths of the sides and is the area of the triangle. This conjecture was shown to be false by Quijano (1915), although it was subsequently proved to be true for acute triangles by Balitrand (1916). A simple counterexample is provided by the triangle with , , and .

Tripolar coordinates

Given a reference triangle and a point , the triple , with , and representing the distances from to the vertices of the reference triangle, is the tripolar coordinates of .The tripolar coordinates satisfy(Euler 1786).Given , the number of points having tripolar coordinates satisfying depends on , and being the sides of a triangle (two points), a degenerate triangle (one point) or not a triangle (zero points) (Bottema 1987)The following table summarizes the tripolar coordinated for a number of named centers.centertripolar coordinatesincenter triangle centroid circumcenter orthocenter symmedian point

Trilinear coordinates

Given a reference triangle , the trilinear coordinates of a point with respect to are an ordered triple of numbers, each of which is proportional to the directed distance from to one of the side lines. Trilinear coordinates are denoted or and also are known as homogeneous coordinates or "trilinears." Trilinear coordinates were introduced by Plücker in 1835. Since it is only the ratio of distances that is significant, the triplet of trilinear coordinates obtained by multiplying a given triplet by any nonzero constant describes the same point, so(1)For simplicity, the three polygon vertices , , and of a triangle are commonly written as , , and , respectively.Trilinear coordinates can be normalized so that they give the actual directed distances from to each of the sides. To perform the normalization, let the point in the above diagram have trilinear coordinates and lie at distances , , and from the sides , , and , respectively. Then..

Barycentric coordinates

Barycentric coordinates are triples of numbers corresponding to masses placed at the vertices of a reference triangle . These masses then determine a point , which is the geometric centroid of the three masses and is identified with coordinates . The vertices of the triangle are given by , , and . Barycentric coordinates were discovered by Möbius in 1827 (Coxeter 1969, p. 217; Fauvel et al. 1993).To find the barycentric coordinates for an arbitrary point , find and from the point at the intersection of the line with the side , and then determine as the mass at that will balance a mass at , thus making the centroid (left figure). Furthermore, the areas of the triangles , , and are proportional to the barycentric coordinates , , and of (right figure; Coxeter 1969, p. 217).Barycentric coordinates are homogeneous, so(1)for .Barycentric coordinates normalized so that they become the actual areas of the subtriangles are called homogeneous..

Homogeneous barycentric coordinates

Homogeneous barycentric coordinates are barycentric coordinates normalized such that they become the actual areas of the subtriangles. Barycentric coordinates normalized so that(1)so that the coordinates give the areas of the subtriangles normalized by the area of the original triangle are called areal coordinates (Coxeter 1969, p. 218). Barycentric and areal coordinates can provide particularly elegant proofs of geometric theorems such as Routh's theorem, Ceva's theorem, and Menelaus' theorem (Coxeter 1969, pp. 219-221).The homogeneous barycentric coordinates corresponding to exact trilinear coordinates are , where(2)(3)(4)The homogeneous barycentric coordinates for some common triangle centers are summarized in the following table, where is the circumradius of the reference triangle.triangle centerhomogeneous barycentric coordinatescircumcenter incenter orthocenter symmedian point triangle..

Areal coordinates

Barycentric coordinates normalized so that they become the areas of the triangles , , and , where is the point whose coordinates have been specified, normalized by the area of the original triangle . This is equivalent to application of the normalization relation(Coxeter 1969, p. 218).

Perpendicular bisector

A perpendicular bisector of a line segment is a line segment perpendicular to and passing through the midpoint of (left figure). The perpendicular bisector of a line segment can be constructed using a compass by drawing circles centered at and with radius and connecting their two intersections. This line segment crosses at the midpoint of (middle figure). If the midpoint is known, then the perpendicular bisector can be constructed by drawing a small auxiliary circle around , then drawing an arc from each endpoint that crosses the line at the farthest intersection of the circle with the line (i.e., arcs with radii and respectively). Connecting the intersections of the arcs then gives the perpendicular bisector (right figure). Note that if the classical construction requirement that compasses be collapsible is dropped, then the auxiliary circle can be omitted and the rigid compass can be used to immediately draw the two arcs using any radius..

Triangle squaring

Let be the altitude of a triangle and let be its midpoint. Thenand can be squared by rectangle squaring. The general polygon can be treated by drawing diagonals, squaring the constituent triangles, and then combining the squares together using the Pythagorean theorem.

Jung's theorem

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).

Rigid circle packing

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).

Circle packing

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"..

Apollonian gasket

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...

Wallpaper groups

The wallpaper groups are the 17 possible plane symmetry groups. They are commonly represented using Hermann-Mauguin-like symbols or in orbifold notation (Zwillinger 1995, p. 260).orbifold notationHermann-Mauguin symbolop12222p2**pmxxpg*2222pmm22*pmg22xpggx*cm2*22cmm442p4*442p4m4*2p4g333p3*333p3ml3*3p3lm632p6*632p6mPatterns created with Artlandia SymmetryWorks for each of these groups are illustrated above.Beautiful patterns can be created by repeating geometric and artistic motifs according to the symmetry of the wallpaper groups, as exemplified in works by M. C. Escher and in the patterns created by I. Bakshee in the Wolfram Language using Artlandia, illustrated above.For a description of the symmetry elements present in each space group, see Coxeter (1969, p. 413)...

Circle division by lines

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.

Pizza theorem

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 :


The word "median" has several different meanings in mathematics all related to the "middle" of mathematical objects.The statistical median is an order statistic that gives the "middle" value of a sample. More specifically, it is the value such that an equal number of samples are less than and greater than the value (for an odd sample size), or the average of the two central values (for an even sample size). The Wolfram Language function Median[list] can be used to find the statistical median of the elements in a list.A triangle median is the Cevian from one of its vertices to the midpoint of the opposite side. The medians intersect in a point known as the triangle centroid that is sometimes also called the median point.Similarly, a tetrahedron median is a line joining a vertex of a tetrahedron to the geometric centroid of the opposite face.The median of a trapezoid is the line segment determinedby the midpoints of..


There are at least two meanings on the word congruent in mathematics. Two geometric figures are said to be congruent if one can be transformed into the other by an isometry (Coxeter and Greitzer 1967, p. 80). This relationship, called geometric congruence, is written . (Unfortunately, the symbol is also used to denote an isomorphism.)A number is said to be congruent to modulo if ( divides ).

Illumination problem

In the early 1950s, Ernst Straus asked 1. Is every region illuminable from every point in the region? 2. Is every region illuminable from at least one point in the region? Here, illuminable means that there is a path from every point to every other by repeated reflections.In 1958, a young Roger Penrose used the properties of the ellipse to describe a room with curved walls that would always have dark (unilluminated) regions, regardless of the position of the candle. Penrose's room, illustrated above, consists of two half-ellipses at the top and bottom and two mushroom-shaped protuberances (which are in turn built up from straight line segments and smaller half-ellipses) on the left and right sides. The ellipses and mushrooms are strategically placed as shown, with the red points being the foci of the half-ellipses. There are essentially three possible configurations of illumination. In this figure, lit regions are indicated in white, unilluminated..

Gauss's circle problem

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..

Linear space

There are at least two distinct notions of linear space throughout mathematics.The term linear space is most commonly used within functionalanalysis as a synonym of the term vector space.The term is also used to describe a fundamental notion in the field of incidence geometry. In particular, a linear space is a space consisting of a collection of points and a set of lines subject to the following axioms: 1. Any two distinct points of belong to exactly one line of . 2. Any line of has at least two points of . 3. There are at least three points of not on a common line. Using this terminology, lines are considered to be "distinguished subsets" of the collection of points. Moreover, in this context, one can view a linear space as a generalization of the notions of projective space and affine space (Batten and Beutelspracher 2009)...

Ternary diagram

A ternary diagram is a triangular diagram which displays the proportion of three variables that sum to a constant and which does so using barycentric coordinates. The coordinate axes of such a diagram are shown in the figure above, where each of the x-, y-, and z-axes are scaled so that , and where the grid lines denote the values , . In most instances, ternary plots are drawn on equilateral triangles as in the figure above, though it is not uncommon for certain scenarios to be better graphed on right triangular diagrams as well (West 2013).Ternary diagrams are sometimes called ternary plots, triangle plots, ternary graphs, simplex plots, and de Finetti diagrams, though the latter term is usually reserved for a specific family of ternary diagrams commonly studied in population genetics. Such diagrams are encountered often in the study of phase equilibria and appear somewhat often throughout a number of physical sciences.pointcoordinates For..

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