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..
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.
When all the points on one line are related by an isometry to all points on another, the midpoints of the segments are either distinct and collinear or coincident.
If three conics pass through two given points and , then the lines joining the other two intersections of each pair of conics are concurrent at a point (Evelyn 1974, p. 15). The converse states that if two conics and meet at four points , , , and , and if and are chords of and , respectively, which meet on , then the six points lie on a conic. The dual of the theorem states that if three conics share two common tangents, then their remaining pairs of common tangents intersect at three collinear points.If the points and are taken as the points at infinity, then the theorem reduces to the theorem that radical lines of three circles are concurrent in a point known as the radical center (Evelyn 1974, p. 15).If two of the points and are taken as the points at infinity, then the theorem becomes that if two circles and pass through two points and on a conic , then the lines determined by the pair of intersections of each circle with the conic are parallel (Evelyn..
A straight line is tangent to a given curve at a point on the curve if the line passes through the point on the curve and has slope , where is the derivative of . This line is called a tangent line, or sometimes simply a tangent.
Two or more lines are said to be concurrent if they intersect in a single point. Two lines concur if their trilinear coordinates satisfy(1)Three lines concur if their trilinearcoordinates satisfy(2)(3)(4)in which case the point is(5)Three lines(6)(7)(8)are concurrent if their coefficients satisfy(9)
Two lines in two-dimensional Euclidean space aresaid to be parallel if they do not intersect.In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. Therefore, parallel lines in three-space lie in a single plane (Kern and Blank 1948, p. 9). Lines in three-space which are not parallel but do not intersect are called skew lines.Two trilinear lines(1)(2)are parallel if(3)(Kimberling 1998, p. 29).
A bitangent is a line that is tangentto a curve at two distinct points.Aa general plane quartic curve has 28 bitangents in the complex projective plane. However, as shown by Plücker (1839), the number of real bitangents of a quartic must be 28, 16, or a number less than 9. Plücker (Plücker 1839, Gray 1982) constructed the first as(correcting the typo of for ) for small and positive. Without mentioning its origin or significance, this curve with is termed the ampersand curve by Cundy and Rowlett (1989, p. 72).As noted by Gray (1982), "the 28 bitangents became, and remain, a topic of delight."Trott (1997) subsequently gave the beautiful symmetric quartic curve with 28 real bitangentswhich is illustrated above.
Sylvester's line problem, known as the Sylvester-Gallai theorem in proved form, states that it is not possible to arrange a finite number of points so that a line through every two of them passes through a third unless they are all on a single line. This problem was proposed by Sylvester (1893), who asked readers to "Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line."Woodall (1893) published a four-line "solution," but an editorial comment following his result pointed out two holes in the argument and sketched another line of enquiry, which is characterized as "equally incomplete, but may be worth notice." However, no correct proof was published at the time (Croft et al. 1991, p. 159), but the problem was revived by Erdős (1943) and correctly solved by Grünwald..
Let a chord of constant length be slid around a smooth, closed, convex curve , and choose a point on the chord which divides it into segments of lengths and . This point will trace out a new closed curve , as illustrated above. Provided certain conditions are met, the area between and is given by , as first shown by Holditch in 1858.The Holditch curve for a circle of radius is another circle which, from the theorem, has radius
is an equireciprocal point if, for every chord of a curve , satisfiesfor some constant . The foci of an ellipse are equichordal points.
An equichordal point is a point for which all the chords of a curve passing through are of the same length. In other words, is an equichordal point if, for every chord of length of the curve , satisfies(1)A function satisfying(2)corresponds to a curve with equichordal point (0, 0) and chord length defined by letting be the polar equation of the half-curve for and then superimposing the polar equation over the same range. The curves illustrated above correspond to polar equations of the form(3)for various values of .Although it long remained an outstanding problem (the equichordal point problem), it is now known that a planar convex region can not have two equichordal points (Rychlik 1997).
In plane geometry, a chord is the line segment joining two points on a curve. The term is often used to describe a line segment whose ends lie on a circle.The term is also used in graph theory, where a cycle chord of a graph cycle is an edge not in whose endpoints lie in .In the above figure, is the radius of the circle, is the chord length, is called the apothem, and the sagitta. The shaded region in the left figure is called a circular sector, and the shaded region in the right figure is called a circular segment.There are a number of interesting theorems about chords of circles. All angles inscribed in a circle and subtended by the same chord are equal. The converse is also true: The locus of all points from which a given segment subtends equal angles is a circle.In the left figure above,(1)(Jurgensen 1963, p. 345). In the right figure above,(2)which is a statement of the fact that the circle power is independent of the choice of the line (Coxeter 1969, p. 81;..
A convex set is centro-symmetric, sometimes also called centrally symmetric, if it has a center that bisects every chord of through .
Given a circle, the apothem is the perpendicular distance from the midpoint of a chord to the circle's center. It is also equal to the radius minus the sagitta ,For a regular polygon, the apothem simply is the distance from the center to a side, i.e., the inradius of the polygon.
A point is said to lie between points and (where , , and are distinct collinear points) if . A number of Euclid's proofs depend on the idea of betweenness without explicit mentioning it.All points on a line segment excluding the endpointslie between the endpoints.Let be a partially ordered set, and let . If , then is said to be between and . If in and there is no that is between and , then covers . Conversely, if covers , then no is between and
Two lines in two-dimensional Euclidean space are said to be parallel if they do not intersect. In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. Lines in three-space that are not parallel but do not intersect are called skew lines.If lines and are parallel, the notation is used.In a non-Euclidean geometry, the concept of parallelism must be modified from its intuitive meaning. This is accomplished by changing the so-called parallel postulate. While this has counterintuitive results, the geometries so defined are still completely self-consistent.In a triangle , a triangle median bisects all segments parallel to a given side (Honsberger 1995, p. 87).
Pairs of points of a line, the product of whose distances from a fixed point is a given constant. This is more concisely defined as a projectivity of period two.If is a range in involution, then the ranges and are equicross, and conversely.
Let , , ... be an infinite series of real numbers lying between 0 and 1. Then corresponding to any arbitrarily large , there exists a positive integer and two subintervals of equal length such that the number of with , 2, ..., which lie in one of the subintervals differs from the number of such that lie in the other subinterval by more than (van der Corput 1935ab, van Aardenne-Ehrenfest 1945, 1949, Roth 1954).This statement can be refined as follows. Let be a large integer and , , ..., be a sequence of real numbers lying between 0 and 1. Then for any integer and any real number satisfying , let denote the number of with , 2, ..., that satisfy . Then there exist and such that(1)where is a positive constant. Schmidt (1972) improved upon this result to obtain(2)which is essentially optimal.This result can be further strengthened, which is most easily done by reformulating the problem. Let be an integer and , , ..., be (not necessarily distinct) points in the square..
Salem constants, sometimes also called Salem numbers, are a set of numbers of which each point of a Pisot number is a limit point from both sides (Salem 1945). The Salem constants are algebraic integers in which one or more of the conjugates is on the unit circle with the others inside (Le Lionnais 1983, p. 150). The smallest known Salem number was found by Lehmer (1933) as the largest real root ofwhich is(OEIS A073011; Le Lionnais 1983, p. 35). This is the famous constant appearing in Lehmer's Mahler measure problem.Boyd (1977) found the following table of small Salem numbers, and suggested that , , , and are the smallest Salem numbers. The notation 1 1 0 is short for 1 1 0 0 1 1, the coefficients of the above polynomial.polynomial11.1762808183101 1 0 21.1883681475181 1 0 0 1 131.2000265240141 0 0 0 0 141.2026167437141 0 0 0 0 0 51.2163916611101 0 0 0 61.2197208590181 0 0 0 0 0 0 171.2303914344101 0 0 0 81.2326135486201 0 0 0 1 0 0 191.2356645804221..
If , , , and are points in the extended complex plane , their cross ratio, also called the cross-ratio (Courant and Robbins 1996, p. 172; Durell 1928, p. 73), anharmonic ratio, and anharmonic section (Casey 1888), is defined by(1)Here if , the result is infinity, and if one of , , , or is infinity, then the two terms on the right containing it are cancelled.For a linear fractional transformation ,(2)The function is the unique linear fractional transformation which takes to 0, to 1, and to infinity. Moreover, is real if and only if the four points lie on a straight line or a generalized circle.There are six different values which the cross ratio may take, depending on the order in which the points are chosen. Let . Possible values of the cross-ratio are then , , , , , and .Given four collinear points , , , and , let the distance between points and be denoted , etc. Then the cross ratio can be defined by(3)The notation is sometimes also used (Coxeter and..
If the cross ratio of satisfy(1)then the points are said to form a bivalent range, and(2)(3)
Given collinear points , , , and , and are harmonic conjugates with respect to and if(1) and are also harmonic conjugates with respect to and .The distances between such points are said to be in a harmonic range, and the line segment depicted above is called a harmonic segment. In other words, harmonic points divide a line segment internally and externally in the same ratio. If , then(2)(3)Harmonic conjugates are also defined for a triangle. If and have trilinear coordinates and , then the trilinear coordinates of the harmonic conjugates are(4)(5)(Kimberling 1994).
A number of points on a line segment. The term was first used by Desargues (Cremona 1960, p. x). If the points , , , ... lie on a line segment with the coordinates of the points such that , they are said to form a range, denoted . Let denote the signed distance . Then the range satisfies the relation(1)The range satisfies(2)and(3)the latter of which holds even when is not on the line (Lachlan 1893).Graustein (1930) and Woods (1961) use the term "range" to refer to the totality of points on a straight line, making it the dual of a pencil.
For signed distances on a line segment,since
The set of all lines through a point. The term was first used by Desargues (Cremona 1960, p. x). The six angles of any pencils of four rays are connected by the relationand the lengths satisfy(Lachlan 1893).Woods (1961) uses the term pencil as a synonym for line segment range, and Altshiller-Court (1979, p. 12) uses the term to mean sheaf of planes.
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,..
A line is a straight one-dimensional figure having no thickness and extending infinitely in both directions. A line is sometimes called a straight line or, more archaically, a right line (Casey 1893), to emphasize that it has no "wiggles" anywhere along its length. While lines are intrinsically one-dimensional objects, they may be embedded in higher dimensional spaces.Harary (1994) called an edge of a graph a "line."A line is uniquely determined by two points, and the line passing through points and is denoted . Similarly, the length of the finite line segment terminating at these points may be denoted . A line may also be denoted with a single lower-case letter (Jurgensen et al. 1963, p. 22).Euclid defined a line as a "breadthless length," and a straight line as a line that "lies evenly with the points on itself" (Kline 1956, Dunham 1990).Consider first lines in a two-dimensional plane. Two..
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)
The intercept form of a line in the Cartesian plane with x-intercept and y-intercept is given by
The standard form of a line in the Cartesian plane is givenbyfor real numbers .This form can be derived from any of the other forms (point-slope form, slope-intercept form, etc.), but can be seen most intuitively when starting from intercept form. Indeed, the intercept form of a line with x-intercept and y-intercept is given byand so by clearing denominators and setting , one gets precisely that .
The 27 real or imaginary lines which lie on the general cubic surface and the 45 triple tangent planes to the surface. All are related to the 28 bitangents of the general quartic curve.Schoute (1910) showed that the 27 lines can be put into a one-to-one correspondence with the vertices of a particular polytope in six-dimensional space in such a manner that all incidence relations between the lines are mirrored in the connectivity of the polytope and conversely (Du Val 1933). A similar correspondence can be made between the 28 bitangents and a seven-dimensional polytope (Coxeter 1928) and between the tritangent planes of the canonical curve of genus four and an eight-dimensional polytope (Du Val 1933).
Two or more lines which have no intersections but are not parallel, also called agonic lines. Since two lines in the plane must intersect or be parallel, skew lines can exist only in three or more dimensions.Two lines with equations(1)(2)are skew if(3)(Gellert et al. 1989, p. 539).This is equivalent to the statement that the vertices of the lines are not coplanar,i.e.,(4)Three skew lines always define a one-sheeted hyperboloid, except in the case where they are all parallel to a single plane but not to each other. In this case, they determine a hyperbolic paraboloid (Hilbert and Cohn-Vossen 1999, p. 15).
A closed interval corresponding to a finite portion of an infinite line. Line segments are generally labeled with two letters corresponding to their endpoints, say and , and then written . The length of the line segment is indicated with an overbar, so the length of the line segment would be written .Curiously, the number of points in a line segment is equal to that in an entire one-dimensional space (a line), and also to the number of points in an -dimensional space, as first recognized by Georg Cantor.
A secant line, also simply called a secant, is a line passing through two points of a curve. As the two points are brought together (or, more precisely, as one is brought towards the other), the secant line tends to a tangent line.The secant line connects two points and in the Cartesian plane on a curve described by a function . It gives the average rate of change of from to (1)which is the slope of the line connecting the points and . The limiting value(2)as the point approaches gives the instantaneous slope of the tangent line to at each point , which is a quantity known as the derivative of , denoted or .The use of secant lines to iteratively find the root of a function is known as thesecant method.In abstract mathematics, the points connected by a secant line can be either realor complex conjugate imaginary.In geometry, a secant line commonly refers to a line that intersects a circle at exactly two points (Rhoad et al. 1984, p. 429). There are a number..
Three or more points , , , ..., are said to be collinear if they lie on a single straight line . A line on which points lie, especially if it is related to a geometric figure such as a triangle, is sometimes called an axis.Two points are trivially collinear since two points determine a line.Three points for , 2, 3 are collinear iff the ratios of distances satisfy(1)A slightly more tractable condition is obtained by noting that the area of a triangle determined by three points will be zero iff they are collinear (including the degenerate cases of two or all three points being concurrent), i.e.,(2)or, in expanded form,(3)This can also be written in vector form as(4)where is the sum of components, , and .The condition for three points , , and to be collinear can also be expressed as the statement that the distance between any one point and the line determined by the other two is zero. In three dimensions, this means setting in the point-line distance(5)giving..
A theorem stated in 1882 which cannot be derived from Euclid's postulates. Given points , , , and on a line, if it is known that the points are ordered as and , then it is also true that .
For triangles in the plane,(1)For spherical triangles,(2)This can be generalized to -gons , where a transversal cuts the side in for , ..., , by(3)Here, and(4)is the ratio of the lengths and with a plus or minus sign depending if these segments have the same or opposite directions (Grünbaum and Shepard 1995). The case is Pasch's axiom.
A geometric theorem related to the pentagram and alsocalled the Pratt-kasapi theorem. It states(1)(2)In general, it is also true that(3)This type of identity was generalized to other figures in the plane and their duals by Pinkernell (1996).
Two distinct point pairs and separate each other if , , , and lie on a circle (or line) in such order that either of the arcs (or the line segment ) contains one but not both of and . In addition, the point pairs separate each other if every circle through and intersects (or coincides with) every circle through and . If the point pairs separate each other, then the symbol is used.
Given a triangle with polygon vertices , , and and points along the sides , , and , a necessary and sufficient condition for the cevians , , and to be concurrent (intersect in a single point) is that(1)This theorem was first published by Giovanni Ceva 1678.Let be an arbitrary -gon, a given point, and a positive integer such that . For , ..., , let be the intersection of the lines and , then(2)Here, and(3)is the ratio of the lengths and with a plus or minus sign depending on whether these segments have the same or opposite directions (Grünbaum and Shepard 1995).Another form of the theorem is that three concurrent lines from the polygon vertices of a triangle divide the opposite sides in such fashion that the product of three nonadjacent segments equals the product of the other three (Johnson 1929, p. 147)...
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,..
A Pisot number is a positive algebraic integer greater than 1 all of whose conjugate elements have absolute value less than 1. A real quadratic algebraic integer greater than 1 and of degree 2 or 3 is a Pisot number if its norm is equal to . The golden ratio (denoted when considered as a Pisot number) is an example of a Pisot number since it has degree two and norm .The smallest Pisot number is given by the positive root (OEIS A060006) of(1)known as the plastic constant. This number was identified as the smallest known by Salem (1944), and proved to be the smallest possible by Siegel (1944).Pisot constants give rise to almost integers. For example, the larger the power to which is taken, the closer , where is the floor function, is to either 0 or 1 (Trott 2004). For example, the spectacular example is within of an integer (Trott 2004, pp. 8-9).The powers of for which this quantity is closer to 0 are 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 17, ... (OEIS A051016), while..
A quantity which gives the inclination of a curve or line with respect to another curve or line. For a line in the -plane making an angle with the x-axis, the slope is a constant given by(1)where and are changes in the two coordinates over some distance.For a plane curve specified as , the slope is(2)for a curve specified parametrically as , the slope is(3)where and , for a curve specified as , the slope is(4)and for a curve given in polar coordinates as , the slope is(5)(Lawrence 1972, pp. 8-9).It is meaningless to talk about the slope of a curve in three-dimensional space unlessthe slope with respect to what is specified.J. Miller has undertaken a detailed study of the origin of the symbol to denote slope. The consensus seems to be that it is not known why the letter was chosen. One high school algebra textbook says the reason for is unknown, but remarks that it is interesting that the French word for "to climb" is "monter."..
The 60 Pascal lines of a hexagon inscribed in a conic intersect three at a time through 20 Steiner points, and also three at a time in 60 Kirkman points. Each Steiner point lies together with three Kirkman points on a total of 20 lines known as Cayley lines. The 20 Cayley lines pass four at a time though 15 points known as Salmon points (Wells 1991). There is a dual relationship between the 20 Cayley lines and the 20 steiner points.
The perpendicular bisector of a linesegment is the locus of all points that are equidistant from its endpoints.This theorem can be applied to determine the center of a given circle with straightedge and compass. Pick three points , and on the circle. Since the center is equidistant from all of them, it lies on the bisector of segment and also on the bisector of segment , i.e., it is the intersection point of the two bisectors. This construction is shown on a window pane by tutor Justin McLeod (Mel Gibson) to his pupil Chuck Norstadt (Nick Stahl) in the 1993 film The Man Without a Face.
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..
Given an arrangement of points, a line containing just two of them is called an ordinary line. Dirac (1951) conjectured that every sufficiently set of noncollinear points contains at least ordinary lines (Borwein and Bailey 2003, p. 18).Csima and Sawyer (1993) proved that for an arrangement of points, at least lines must be ordinary. Only two exceptions are known for Dirac's conjecture: the Kelly-Moser configuration (7 points, 3 ordinary lines; cf. Fano plane) and McKee's configuration (13 points, 6 ordinary lines).Silva and Fukuda conjectured that for any noncollinear, equally distributed, line-separable arrangement of points of two colors, there is at least one bichromatic ordinary line. Finschi and Fukuda found a unique nine-point counterexample in a study of 15296266 distinct configurations (Malkevitch)...
Archimedes' axiom, also known as the continuity axiom or Archimedes' lemma, survives in the writings of Eudoxus (Boyer and Merzbach 1991), but the term was first coined by the Austrian mathematician Otto Stolz (1883). It states that, given two magnitudes having a ratio, one can find a multiple of either which will exceed the other. This principle was the basis for the method of exhaustion, which Archimedes invented to solve problems of area and volume.Symbolically, the axiom states thatiff the appropriate one of following conditions is satisfied for integers and : 1. If , then . 2. If , then . 3. If , then . Formally, Archimedes' axiom states that if and are two line segments, then there exist a finite number of points , , ..., on such thatand is between and (Itô 1986, p. 611). A geometry in which Archimedes' lemma does not hold is called a non-Archimedean Geometry...