In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices; the interior surface of such a polygon is not uniquely defined. Skew infinite polygons have vertices. A zig-zag skew polygon or antiprismatic polygon has vertices which alternate on two parallel planes, thus must be even-sided. Regular skew polygon in 3 dimensions are always zig-zag. A regular skew polygon is isogonal with equal edge lengths. In 3 dimensions a regular skew polygon is a zig-zag skew, with vertices alternating between two parallel planes; the sides of an n-antiprism can define a regular skew 2n-gons. A regular skew n-gonal can be given a symbol # as a blend of a regular polygon, an orthogonal line segment; the symmetry operation between sequential vertices is glide reflection. Examples are shown on pentagon antiprisms; the star antiprisms generate regular skew polygons with different connection order of the top and bottom polygons. The filled top and bottom polygons are drawn for structural clarity, are not part of the skew polygons.
A regular compound skew 2n-gon can be constructed by adding a second skew polygon by a rotation. These shares the same vertices as the prismatic compound of antiprisms. Petrie polygons are regular skew polygons defined within regular polyhedra and polytopes. For example, the 5 Platonic solids have 4-, 6-, 10-sided regular skew polygons, as seen in these orthogonal projections with red edges around the projective envelope; the tetrahedron and octahedron include all the vertices in the zig-zag skew polygon and can be seen as a digonal and a triangular antiprisms respectively. The regular skew polyhedron have regular faces, regular skew polygon vertex figures. Three are infinite space-filling in 3-space and others exist in 4-space, some within the uniform 4-polytope. An isogonal skew polygon is a skew polygon with one type of vertex, connected by two types of edges. Isogonal skew polygons with equal edge lengths can be considered quasiregular, it is similar to a zig-zag skew polygon, existing on two planes, except allowing one edge to cross to the opposite plane, the other edge to stay on the same plane.
Isogonal skew polygons can be defined on even-sided n-gonal prisms, alternatingly following an edge of one side polygon, moving between polygons. For example, on the vertices of a cube. Vertices alternate between top and bottom squares with red edges between sides, blue edges along each side. In 4 dimensions a regular skew polygon can have vertices on a Clifford torus and related by a Clifford displacement. Unlike zig-zag skew polygons, skew polygons on double rotations can include an odd-number of sides; the petrie polygons of the regular 4-polytope define regular skew polygons. The Coxeter number for each coxeter group symmetry expresses; this is 5 sides for a 5-cell, 8 sides for a tesseract and 16-cell, 12 sides for a 24-cell, 30 sides for a 120-cell and 600-cell. When orthogonally projected onto the Coxeter plane these regular skew polygons appear as regular polygon envelopes in the plane; the n-n duoprism and dual duopyramids have 2n-gonal petrie polygons. Petrie polygon Quadrilateral#Skew quadrilaterals Regular skew polyhedron Skew apeirohedron Skew lines McMullen, Peter.
The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. "Skew Polygons" §2.2 Coxeter, H. S. M.. Chapter 1. Regular polygons, 1.5. Regular polygons in n dimensions, 1.7. Zigzag and antiprismatic polygons, 1.8. Helical polygons. 4.3. Flags and Orthoschemes, 11.3. Petrie polygons H. S. M. Petrie Polygons. Regular Polytopes, 3rd ed. New York: Dover, 1973. Coxeter, H. S. M. & Moser, W. O. J.. Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9. 5.2 The Petrie polygon. John Milnor: On the total curvature of knots, Ann. Math. 52 248–257. J. M. Sullivan: Curves of finite total curvature, ArXiv:math.0606007v2 Weisstein, Eric W. "Skew polygon". MathWorld. Weisstein, Eric W. "Petrie polygon". MathWorld
Arthur Cayley was a British mathematician. He helped; as a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, where he excelled in Greek, French and Italian, as well as mathematics, he worked as a lawyer for 14 years. He postulated the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, verified it for matrices of order 2 and 3, he was the first to define the concept of a group in the modern way—as a set with a binary operation satisfying certain laws. When mathematicians spoke of "groups", they had meant permutation groups. Cayley tables and Cayley graphs as well. Arthur Cayley was born in Richmond, England, on 16 August 1821, his father, Henry Cayley, was a distant cousin of Sir George Cayley, the aeronautics engineer innovator, descended from an ancient Yorkshire family. He settled in Russia, as a merchant, his mother was daughter of William Doughty. According to some writers she was Russian, his brother was the linguist Charles Bagot Cayley.
Arthur spent his first eight years in Saint Petersburg. In 1829 his parents were settled permanently near London. Arthur was sent to a private school. At age 14 he was sent to King's College School; the school's master observed indications of mathematical genius and advised the father to educate his son not for his own business, as he had intended, but to enter the University of Cambridge. At the unusually early age of 17 Cayley began residence at Cambridge; the cause of the Analytical Society had now triumphed, the Cambridge Mathematical Journal had been instituted by Gregory and Robert Leslie Ellis. To this journal, at the age of twenty, Cayley contributed three papers, on subjects, suggested by reading the Mécanique analytique of Lagrange and some of the works of Laplace. Cayley's tutor at Cambridge was George Peacock and his private coach was William Hopkins, he finished his undergraduate course by winning the place of Senior Wrangler, the first Smith's prize. His next step was to take the M.
A. degree, win a Fellowship by competitive examination. He continued to reside at Cambridge University for four years; because of the limited tenure of his fellowship it was necessary to choose a profession. He made a specialty of conveyancing, it was while he was a pupil at the bar examination that he went to Dublin to hear Hamilton's lectures on quaternions. His friend J. J. Sylvester, his senior by five years at Cambridge, was an actuary, resident in London. During this period of his life, extending over fourteen years, Cayley produced between two and three hundred papers. At Cambridge University the ancient professorship of pure mathematics is denominated by the Lucasian, is the chair, occupied by Isaac Newton. Around 1860, certain funds bequeathed by Lady Sadleir to the University, having become useless for their original purpose, were employed to establish another professorship of pure mathematics, called the Sadleirian; the duties of the new professor were defined to be "to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science."
To this chair Cayley was elected. He gave up a lucrative practice for a modest salary, he at once settled down in Cambridge. More fortunate than Hamilton in his choice, his home life was one of great happiness, his friend and fellow investigator, once remarked that Cayley had been much more fortunate than himself. At first the teaching duty of the Sadleirian professorship was limited to a course of lectures extending over one of the terms of the academic year. For many years the attendance was small, came entirely from those who had finished their career of preparation for competitive examinations; the subject lectured on was that of the memoir on which the professor was for the time engaged. The other duty of the chair — the advancement of mathematical science — was discharged in a handsome manner by the long series of memoirs that he published, ranging over every department of pure mathematics, but it was discharged in a much less obtrusive way. In 1872 he was made an honorary fellow of Trinity College, three years an ordinary fellow, which meant stipend as well as honour.
About this time his friends subscribed for a presentation portrait. Maxwell wrote an address to the committee of subscribers; the verses refer to the subjects investigated in several of Cay
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every consecutive sides belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the projection interior to it; the plane in question is the Coxeter plane of the symmetry group of the polygon, the number of sides, h, is Coxeter number of the Coxeter group. These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional regular polytopes. Petrie polygons can be defined more for any embedded graph, they form the faces of another embedding of the same graph on a different surface, called the Petrie dual. John Flinders Petrie was the only son of Egyptologist Flinders Petrie, he as a schoolboy showed remarkable promise of mathematical ability. In periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them.
He first noted the importance of the regular skew polygons which appear on the surface of regular polyhedra and higher polytopes. Coxeter explained in 1937 how he and Petrie began to expand the classical subject of regular polyhedra: One day in 1926, J. F. Petrie told me with much excitement that he had discovered two new regular polyhedral; when my incredulity had begun to subside, he described them to me: one consisting of squares, six at each vertex, one consisting of hexagons, four at each vertex. In 1938 Petrie collaborated with Coxeter, Patrick du Val, H. T. Flather to produce The Fifty-Nine Icosahedra for publication. Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes; the idea of Petrie polygons was extended to semiregular polytopes. The Petrie polygon of the regular polyhedron has h sides, where h + 2 = 24/; the regular duals, are contained within the same projected Petrie polygon. The regular Kepler–Poinsot polyhedra have hexagonal, decagrammic, Petrie polygons.
Infinite regular skew polygons can be defined as being the Petrie polygons of the regular tilings, having angles of 90, 120, 60 degrees of their square and triangular faces respectively. Infinite regular skew polygons exist as Petrie polygons of the regular hyperbolic tilings, like the order-7 triangular tiling,: The Petrie polygon for the regular polychora can be determined; the Petrie polygon projections are most useful for visualization of polytopes of dimension four and higher. This table represents Petrie polygon projections of 3 regular families, the exceptional Lie group En which generate semiregular and uniform polytopes for dimensions 4 to 8. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Coxeter, H. S. M. Regular complex polytopes. Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons Ball, W. W. R. and H. S. M. Coxeter Mathematical Recreations and Essays, 13th ed. New York: Dover. Coxeter, H. S. M; the Beauty of Geometry: Twelve Essays, Dover Publications LCCN 99-35678 Peter McMullen, Egon Schulte Abstract Regular Polytopes, Cambridge University Press.
ISBN 0-521-81496-0 Steinberg, Robert,ON THE NUMBER OF SIDES OF A PETRIE POLYGON Weisstein, Eric W. "Petrie polygon". MathWorld. Weisstein, Eric W. "Hypercube graphs". MathWorld. Weisstein, Eric W. "Cross polytope graphs". MathWorld. Weisstein, Eric W. "24-cell graph". MathWorld. Weisstein, Eric W. "120-cell graph". MathWorld. Weisstein, Eric W. "600-cell graph". MathWorld. Weisstein, Eric W. "Gosset graph 3_21". MathWorld
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.
R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.
The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.
When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form
Small stellated dodecahedron
In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, with Schläfli symbol. It is one of four nonconvex regular polyhedra, it is composed with five pentagrams meeting at each vertex. It shares the same vertex arrangement as the convex regular icosahedron, it shares the same edge arrangement with the great icosahedron, with which it forms a degenerate uniform compound figure. It is the second of four stellations of the dodecahedron; the small stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the edges of the core polytope until a point is reached where they intersect. If the pentagrammic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron, but with much taller isosceles triangle faces, with the height of the pentagonal pyramids adjusted so that the five triangles in the pentagram become coplanar; the critical angle is atan above the dodecahedron face.
If we regard it as having 12 pentagrams as faces, with these pentagrams meeting at 30 edges and 12 vertices, we can compute its genus using Euler's formula V − E + F = 2 − 2 g and conclude that the small stellated dodecahedron has genus 4. This observation, made by Louis Poinsot, was confusing, but Felix Klein showed in 1877 that the small stellated dodecahedron could be seen as a branched covering of the Riemann sphere by a Riemann surface of genus 4, with branch points at the center of each pentagram. In fact this Riemann surface, called Bring's curve, has the greatest number of symmetries of any Riemann surface of genus 4: the symmetric group S 5 acts as automorphisms It can be seen in a floor mosaic in St Mark's Basilica, Venice by Paolo Uccello circa 1430, it is central to two lithographs by M. C. Escher: Contrast and Gravitation, its convex hull is the regular convex icosahedron. It shares its edges with the great icosahedron. There are four related uniform polyhedra, constructed as degrees of truncation.
The dual is a great dodecahedron. The dodecadodecahedron is a rectification; the truncated small stellated dodecahedron can be considered a degenerate uniform polyhedron since edges and vertices coincide, but it is included for completeness. Visually, it looks like a regular dodecahedron on the surface, but it has 24 faces in overlapping pairs; the spikes are truncated. The 24 faces are 12 pentagons from the truncated vertices and 12 decagons taking the form of doubly-wound pentagons overlapping the first 12 pentagons; the latter faces are formed by truncating the original pentagrams. When an -gon is truncated, it becomes a -gon. For example, a truncated pentagon becomes a decagon, so truncating a pentagram becomes a doubly-wound pentagon. Compound of small stellated dodecahedron and great dodecahedron Wenninger, Magnus. Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. Weber, Matthias, "Kepler's small stellated dodecahedron as a Riemann surface", Pacific J. Math. 220: 167–182, doi:10.2140/pjm.2005.220.167 Eric W. Weisstein, Small stellated dodecahedron at MathWorld.
Weisstein, Eric W. "DodecahedronStellations". MathWorld. Uniform polyhedra and duals
In geometry, the midsphere or intersphere of a polyhedron is a sphere, tangent to every edge of the polyhedron. That is to say, it touches any given edge at one point. Not every polyhedron has a midsphere, but for every polyhedron there is a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere; the midsphere is so-called because, for polyhedra that have a midsphere, an inscribed sphere and a circumscribed sphere, the midsphere is in the middle, between the other two spheres. The radius of the midsphere is called the midradius; the uniform polyhedra, including the regular and semiregular polyhedra and their duals all have midspheres. In the regular polyhedra, the inscribed sphere and circumscribed sphere all exist and are concentric. If O is the midsphere of a polyhedron P the intersection of O with any face of P is a circle; the circles formed in this way on all of the faces of P form a system of circles on O that are tangent when the faces they lie in share an edge.
Dually, if v is a vertex of P there is a cone that has its apex at v and, tangent to O in a circle. That is, the circle is the horizon of the midsphere, as viewed from the vertex; the circles formed in this way are tangent to each other when the vertices they correspond to are connected by an edge. If a polyhedron P has a midsphere O the polar polyhedron with respect to O has O as its midsphere; the face planes of the polar polyhedron pass through the circles on O that are tangent to cones having the vertices of P as their apexes. One stronger form of the circle packing theorem, on representing planar graphs by systems of tangent circles, states that every polyhedral graph can be represented by a polyhedron with a midsphere; the horizon circles of a canonical polyhedron can be transformed, by stereographic projection, into a collection of circles in the Euclidean plane that do not cross each other and are tangent to each other when the vertices they correspond to are adjacent. In contrast, there exist polyhedra that do not have an equivalent form with an inscribed sphere or circumscribed sphere.
Any two polyhedra with the same face lattice and the same midsphere can be transformed into each other by a projective transformation of three-dimensional space that leaves the midsphere in the same position. The restriction of this projective transformation to the midsphere is a Möbius transformation. There is a unique way of performing this transformation so that the midsphere is the unit sphere and so that the centroid of the points of tangency is at the center of the sphere. Alternatively, a transformed polyhedron that maximizes the minimum distance of a vertex from the midsphere can be found in linear time. Bern, M.. "Optimal Möbius transformations for information visualization and meshing", 7th Worksh. Algorithms and Data Structures, Lecture Notes in Computer Science, 2125, Rhode Island: Springer-Verlag, pp. 14–25, arXiv:cs. CG/0101006, doi:10.1007/3-540-44634-6_3. Coxeter, H. S. M. "2.1 Regular polyhedra. Cundy, H. M.. Mathematical Models, Oxford University Press, p. 117. Koebe, Paul, "Kontaktprobleme der Konformen Abbildung", Ber.
Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88: 141–164. Sachs, Horst, "Coin graphs and conformal mapping", Discrete Mathematics, 134: 133–138, doi:10.1016/0012-365XE0068-F, MR 1303402. Schramm, Oded, "How to cage an egg", Inventiones Mathematicae, 107: 543–560, Bibcode:1992InMat.107..543S, doi:10.1007/BF01231901, MR 1150601. Steinitz, E. "Über isoperimetrische Probleme bei konvexen Polyedern", Journal für die reine und angewandte Mathematik, 159: 133–143. Ziegler, Günter M. Lectures on Polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, pp. 117–118, ISBN 0-387-94365-X. Hart, G. W. "Calculating canonical polyhedra", Mathematica in Education and Research, 6: 5–10. A Mathematica implementation of an algorithm for constructing canonical polyhedra. Weisstein, Eric W. "Midsphere". MathWorld
In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having two sides of equal length, sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, the faces of bipyramids and certain Catalan solids; the mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from earlier times, appear in architecture and design, for instance in the pediments and gables of buildings; the two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height and perimeter, can be calculated by simple formulas from the lengths of the legs and base; every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base.
The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs. Euclid defined an isosceles triangle as a triangle with two equal sides, but modern treatments prefer to define isosceles triangles as having at least two equal sides; the difference between these two definitions is that the modern version makes equilateral triangles a special case of isosceles triangles. A triangle, not isosceles is called scalene. "Isosceles" is a compound word, made from the Greek roots "isos" and "skelos". The same word is used, for instance, for isosceles trapezoids, trapezoids with two equal sides, for isosceles sets, sets of points every three of which form an isosceles triangle. In an isosceles triangle that has two equal sides, the equal sides are called legs and the third side is called the base; the angle included by the legs is called the vertex angle and the angles that have the base as one of their sides are called the base angles.
The vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base. Whether an isosceles triangle is acute, right or obtuse depends only on the angle at its apex. In Euclidean geometry, the base angles cannot be obtuse or right because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. Since a triangle is obtuse or right if and only if one of its angles is obtuse or right an isosceles triangle is obtuse, right or acute if and only if its apex angle is obtuse, right or acute. In Edwin Abbott's book Flatland, this classification of shapes was used as a satire of social hierarchy: isosceles triangles represented the working class, with acute isosceles triangles higher in the hierarchy than right or obtuse isosceles triangles; as well as the isosceles right triangle, several other specific shapes of isosceles triangles have been studied. These include the Calabi triangle, the golden triangle and golden gnomon, the 80-80-20 triangle appearing in the Langley’s Adventitious Angles puzzle, the 30-30-120 triangle of the triakis triangular tiling.
Five Catalan solids, the triakis tetrahedron, triakis octahedron, tetrakis hexahedron, pentakis dodecahedron, triakis icosahedron, each have isosceles-triangle faces, as do infinitely many pyramids and bipyramids. For any isosceles triangle, the following six line segments coincide: the altitude, a line segment from the apex perpendicular to the base, the angle bisector from the apex to the base, the median from the apex to the midpoint of the base, the perpendicular bisector of the base within the triangle, the segment within the triangle of the unique axis of symmetry of the triangle, the segment within the triangle of the Euler line of the triangle, their common length is the height h of the triangle. If the triangle has equal sides of length a and base of length b, the general triangle formulas for the lengths of these segments all simplify to h = 1 2 4 a 2 − b 2; this formula can be derived from the Pythagorean theorem using the fact that the altitude bisects the base and partitions the isosceles triangle into two congruent right triangles.
The Euler line of any triangle goes through the triangle's orthocenter, its centroid, its circumcenter. In an isosceles triangle with two equal sides, these three points are distinct, all lie on the symmetry axis of the triangle, from which it follows that the Euler line coincides with the axis of symmetry; the incenter of the triangle lies on the Euler line, something, not true for other triangles. If any two of an angle bisector, median, or altitude coincide in a given triangle, that triangle must be isosceles; the area T of an isosceles triangle can be derived from the formula for its height, from the general formula for the area of a triangle as half the product of base and height: T =