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70-484: The bridge extends 186 m and its width is 9 meters wide in an eastern-western direction. The supports of the bridge are hexagonal , and their interior parts are constructed with rubble-stone and plaster mortar , and their facing is of chiseled stone -blocks. The bridge has six spans. The spans have ribbed arches and are made of brick . In the northern and southern sides of the bridge there are brick buttresses on some break waters, and between every two buttresses on

140-472: A : c a c + a − b : a b a + b − c . {\displaystyle {\frac {bc}{b+c-a}}:{\frac {ca}{c+a-b}}:{\frac {ab}{a+b-c}}.} An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two . Every triangle has three distinct excircles, each tangent to one of

210-508: A , y a ) {\displaystyle (x_{a},y_{a})} , ( x b , y b ) {\displaystyle (x_{b},y_{b})} , and ( x c , y c ) {\displaystyle (x_{c},y_{c})} , and the sides opposite these vertices have corresponding lengths a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} , then

280-457: A {\displaystyle a} be the length of B C ¯ {\displaystyle {\overline {BC}}} , b {\displaystyle b} the length of A C ¯ {\displaystyle {\overline {AC}}} , and c {\displaystyle c} the length of A B ¯ {\displaystyle {\overline {AB}}} . Now,

350-658: A {\displaystyle a} be the length of B C ¯ {\displaystyle {\overline {BC}}} , b {\displaystyle b} the length of A C ¯ {\displaystyle {\overline {AC}}} , and c {\displaystyle c} the length of A B ¯ {\displaystyle {\overline {AB}}} . Also let T A {\displaystyle T_{A}} , T B {\displaystyle T_{B}} , and T C {\displaystyle T_{C}} be

420-480: A {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are the side lengths of the original triangle. This is the same area as that of the extouch triangle . The three lines A T A {\displaystyle AT_{A}} , B T B {\displaystyle BT_{B}} and C T C {\displaystyle CT_{C}} intersect in

490-534: A ) = s − a . {\displaystyle d\left(A,T_{B}\right)=d\left(A,T_{C}\right)={\tfrac {1}{2}}(b+c-a)=s-a.} If the altitudes from sides of lengths a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are h a {\displaystyle h_{a}} , h b {\displaystyle h_{b}} , and h c {\displaystyle h_{c}} , then

560-492: A + b + c . {\displaystyle \left({\frac {ax_{a}+bx_{b}+cx_{c}}{a+b+c}},{\frac {ay_{a}+by_{b}+cy_{c}}{a+b+c}}\right)={\frac {a\left(x_{a},y_{a}\right)+b\left(x_{b},y_{b}\right)+c\left(x_{c},y_{c}\right)}{a+b+c}}.} The inradius r {\displaystyle r} of the incircle in a triangle with sides of length a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c}

630-732: A r {\displaystyle {\tfrac {1}{2}}ar} . Since these three triangles decompose △ A B C {\displaystyle \triangle ABC} , we see that the area Δ  of  △ A B C {\displaystyle \Delta {\text{ of }}\triangle ABC} is: Δ = 1 2 ( a + b + c ) r = s r , {\displaystyle \Delta ={\tfrac {1}{2}}(a+b+c)r=sr,}      and      r = Δ s , {\displaystyle r={\frac {\Delta }{s}},} where Δ {\displaystyle \Delta }

700-409: A hexagram . A regular hexagon can be dissected into six equilateral triangles by adding a center point. This pattern repeats within the regular triangular tiling . A regular hexagon can be extended into a regular dodecagon by adding alternating squares and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling . There are six self-crossing hexagons with

770-446: A triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter . An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two . Every triangle has three distinct excircles, each tangent to one of

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840-423: A hexagon has vertices on the circumcircle of an acute triangle at the six points (including three triangle vertices) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle. Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at

910-403: A point in the triangle is the ratio of all the distances to the triangle sides. Because the incenter is the same distance from all sides of the triangle, the trilinear coordinates for the incenter are   1 : 1 : 1. {\displaystyle \ 1:1:1.} The barycentric coordinates for a point in a triangle give weights such that the point is the weighted average of

980-434: A regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then PE + PF = PA + PB + PC + PD . It follows from the ratio of circumradius to inradius that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long diagonal of 1.0000000 will have a distance of 0.8660254 or cos(30°) between parallel sides. For an arbitrary point in

1050-575: A regular hexagonal pattern. The two simple roots have a 120° angle between them. The 12 roots of the Exceptional Lie group G2 , represented by a Dynkin diagram [REDACTED] [REDACTED] [REDACTED] are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them. Coxeter states that every zonogon (a 2 m -gon whose opposite sides are parallel and of equal length) can be dissected into 1 ⁄ 2 m ( m − 1) parallelograms. In particular this

1120-406: A right-angled triangle with one side equal to r {\displaystyle r} and the other side equal to r cot ⁡ A 2 {\displaystyle r\cot {\tfrac {A}{2}}} . The same is true for △ I B ′ A {\displaystyle \triangle IB'A} . The large triangle is composed of six such triangles and

1190-543: A single point called the Gergonne point , denoted as G e {\displaystyle G_{e}} (or triangle center X 7 ). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and can be any point therein. The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle. Trilinear coordinates for

1260-423: A single point. In a hexagon that is tangential to a circle and that has consecutive sides a , b , c , d , e , and f , If an equilateral triangle is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the centroids of opposite triangles form another equilateral triangle. A skew hexagon is a skew polygon with six vertices and edges but not existing on

1330-434: A triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. The incircle radius is no greater than one-ninth the sum of the altitudes. The squared distance from the incenter I {\displaystyle I} to the circumcenter O {\displaystyle O}

1400-398: Is K T = K 2 r 2 s a b c {\displaystyle K_{T}=K{\frac {2r^{2}s}{abc}}} where K {\displaystyle K} , r {\displaystyle r} , and s {\displaystyle s} are the area, radius of the incircle , and semiperimeter of the original triangle, and

1470-409: Is 720°. A regular hexagon has Schläfli symbol {6} and can also be constructed as a truncated equilateral triangle , t{3}, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral and equiangular . It is bicentric , meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of

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1540-536: Is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite A {\displaystyle A} is denoted T A {\displaystyle T_{A}} , etc. This Gergonne triangle, △ T A T B T C {\displaystyle \triangle T_{A}T_{B}T_{C}} , is also known as the contact triangle or intouch triangle of △ A B C {\displaystyle \triangle ABC} . Its area

1610-750: Is full symmetry, and a1 is no symmetry. p6 , an isogonal hexagon constructed by three mirrors can alternate long and short edges, and d6 , an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an elongated rhombus , while d2 and p2 can be seen as horizontally and vertically elongated kites . g2 hexagons, with opposite sides parallel are also called hexagonal parallelogons . Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only

1680-549: Is given by O I ¯ 2 = R ( R − 2 r ) = a b c a + b + c [ a b c ( a + b − c ) ( a − b + c ) ( − a + b + c ) − 1 ] {\displaystyle {\overline {OI}}^{2}=R(R-2r)={\frac {a\,b\,c\,}{a+b+c}}\left[{\frac {a\,b\,c\,}{(a+b-c)\,(a-b+c)\,(-a+b+c)}}-1\right]} and

1750-411: Is given by r = ( s − a ) ( s − b ) ( s − c ) s , {\displaystyle r={\sqrt {\frac {(s-a)(s-b)(s-c)}{s}}},} where s = 1 2 ( a + b + c ) {\displaystyle s={\tfrac {1}{2}}(a+b+c)} is the semiperimeter. The tangency points of

1820-439: Is inscribed in any conic section , and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration. The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point . If

1890-447: Is no Platonic solid made of only regular hexagons, because the hexagons tessellate , not allowing the result to "fold up". The Archimedean solids with some hexagonal faces are the truncated tetrahedron , truncated octahedron , truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron . These hexagons can be considered truncated triangles, with Coxeter diagrams of

1960-496: Is the Petrie polygon for these higher dimensional regular , uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections : A principal diagonal of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a , there exists a principal diagonal d 1 such that and a principal diagonal d 2 such that There

2030-400: Is the area of △ A B C {\displaystyle \triangle ABC} and s = 1 2 ( a + b + c ) {\displaystyle s={\tfrac {1}{2}}(a+b+c)} is its semiperimeter . For an alternative formula, consider △ I T C A {\displaystyle \triangle IT_{C}A} . This is

2100-406: Is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a Petrie polygon projection of a cube , with 3 of 6 square faces. Other parallelogons and projective directions of the cube are dissected within rectangular cuboids . A regular hexagon has Schläfli symbol {6}. A regular hexagon is a part of

2170-1027: The Law of sines in the triangle △ I A B {\displaystyle \triangle IAB} . We get A I ¯ sin ⁡ B 2 = c sin ⁡ ∠ A I B {\displaystyle {\frac {\overline {AI}}{\sin {\frac {B}{2}}}}={\frac {c}{\sin \angle AIB}}} . We have that ∠ A I B = π − A 2 − B 2 = π 2 + C 2 {\displaystyle \angle AIB=\pi -{\frac {A}{2}}-{\frac {B}{2}}={\frac {\pi }{2}}+{\frac {C}{2}}} . It follows that A I ¯ = c   sin ⁡ B 2 cos ⁡ C 2 {\displaystyle {\overline {AI}}=c\ {\frac {\sin {\frac {B}{2}}}{\cos {\frac {C}{2}}}}} . The equality with

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2240-484: The dihedral group D 6 . The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral , and that the regular hexagon can be partitioned into six equilateral triangles. Like squares and equilateral triangles , regular hexagons fit together without any gaps to tile

2310-451: The excenter of A . Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system . Suppose △ A B C {\displaystyle \triangle ABC} has an incircle with radius r {\displaystyle r} and center I {\displaystyle I} . Let

2380-462: The g6 subgroup has no degrees of freedom but can be seen as directed edges . Hexagons of symmetry g2 , i4 , and r12 , as parallelogons can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations. The 6 roots of the simple Lie group A2 , represented by a Dynkin diagram [REDACTED] [REDACTED] [REDACTED] , are in

2450-542: The vertex arrangement of the regular hexagon: From bees' honeycombs to the Giant's Causeway , hexagonal patterns are prevalent in nature due to their efficiency. In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest hexagons. This means that honeycombs require less wax to construct and gain much strength under compression . Irregular hexagons with parallel opposite edges are called parallelogons and can also tile

2520-424: The Gergonne point are given by sec 2 ⁡ A 2 : sec 2 ⁡ B 2 : sec 2 ⁡ C 2 , {\displaystyle \sec ^{2}{\tfrac {A}{2}}:\sec ^{2}{\tfrac {B}{2}}:\sec ^{2}{\tfrac {C}{2}},} or, equivalently, by the Law of Sines , b c b + c −

2590-445: The angles at the three vertices. The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter (that is, using the barycentric coordinates given above, normalized to sum to unity) as weights. The weights are positive so the incenter lies inside the triangle as stated above. If the three vertices are located at ( x

2660-494: The area can also be expressed in terms of the apothem a and the perimeter p . For the regular hexagon these are given by a = r , and p = 6 R = 4 r 3 {\displaystyle {}=6R=4r{\sqrt {3}}} , so The regular hexagon fills the fraction 3 3 2 π ≈ 0.8270 {\displaystyle {\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270} of its circumscribed circle . If

2730-425: The distance from the incenter to the center N {\displaystyle N} of the nine point circle is I N ¯ = 1 2 ( R − 2 r ) < 1 2 R . {\displaystyle {\overline {IN}}={\tfrac {1}{2}}(R-2r)<{\tfrac {1}{2}}R.} The incenter lies in the medial triangle (whose vertices are

2800-546: The etymology of the term. The prefix "hex-" originates from the Greek word "hex," meaning six, while "sex-" comes from the Latin "sex," also signifying six. Some linguists and mathematicians argue that since many English mathematical terms derive from Latin, the use of "sexagon" would align with this tradition. Historical discussions date back to the 19th century, when mathematicians began to standardize terminology in geometry. However,

2870-457: The excircles are called the exradii . The exradius of the excircle opposite A {\displaystyle A} (so touching B C {\displaystyle BC} , centered at J A {\displaystyle J_{A}} ) is r a = r s s − a = s ( s − b ) ( s − c ) s −

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2940-440: The form [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] and [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] . There are other symmetry polyhedra with stretched or flattened hexagons, like these Goldberg polyhedron G(2,0): There are also 9 Johnson solids with regular hexagons: The debate over whether hexagons should be referred to as "sexagons" has its roots in

3010-445: The hexagon), D , is twice the maximal radius or circumradius , R , which equals the side length, t . The minimal diameter or the diameter of the inscribed circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), d , is twice the minimal radius or inradius , r . The maxima and minima are related by the same factor: The area of a regular hexagon For any regular polygon ,

3080-500: The incenter is at ( a x a + b x b + c x c a + b + c , a y a + b y b + c y c a + b + c ) = a ( x a , y a ) + b ( x b , y b ) + c ( x c , y c )

3150-745: The incenter of △ A B C {\displaystyle \triangle ABC} as I {\displaystyle I} . The distance from vertex A {\displaystyle A} to the incenter I {\displaystyle I} is: A I ¯ = d ( A , I ) = c sin ⁡ B 2 cos ⁡ C 2 = b sin ⁡ C 2 cos ⁡ B 2 . {\displaystyle {\overline {AI}}=d(A,I)=c\,{\frac {\sin {\frac {B}{2}}}{\cos {\frac {C}{2}}}}=b\,{\frac {\sin {\frac {C}{2}}}{\cos {\frac {B}{2}}}}.} Use

3220-447: The incircle divide the sides into segments of lengths s − a {\displaystyle s-a} from A {\displaystyle A} , s − b {\displaystyle s-b} from B {\displaystyle B} , and s − c {\displaystyle s-c} from C {\displaystyle C} . See Heron's formula . Denote

3290-1121: The incircle is tangent to A B ¯ {\displaystyle {\overline {AB}}} at some point T C {\displaystyle T_{C}} , and so ∠ A T C I {\displaystyle \angle AT_{C}I} is right. Thus, the radius T C I {\displaystyle T_{C}I} is an altitude of △ I A B {\displaystyle \triangle IAB} . Therefore, △ I A B {\displaystyle \triangle IAB} has base length c {\displaystyle c} and height r {\displaystyle r} , and so has area 1 2 c r {\displaystyle {\tfrac {1}{2}}cr} . Similarly, △ I A C {\displaystyle \triangle IAC} has area 1 2 b r {\displaystyle {\tfrac {1}{2}}br} and △ I B C {\displaystyle \triangle IBC} has area 1 2

3360-494: The incircle radius r {\displaystyle r} and the circumcircle radius R {\displaystyle R} of a triangle with sides a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} is r R = a b c 2 ( a + b + c ) . {\displaystyle rR={\frac {abc}{2(a+b+c)}}.} Some relations among

3430-402: The inradius r {\displaystyle r} is one-third of the harmonic mean of these altitudes; that is, r = 1 1 h a + 1 h b + 1 h c . {\displaystyle r={\frac {1}{{\dfrac {1}{h_{a}}}+{\dfrac {1}{h_{b}}}+{\dfrac {1}{h_{c}}}}}.} The product of

3500-908: The internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system . While the incenter of △ A B C {\displaystyle \triangle ABC} has trilinear coordinates 1 : 1 : 1 {\displaystyle 1:1:1} , the excenters have trilinears J A = − 1 : 1 : 1 J B = 1 : − 1 : 1 J C = 1 : 1 : − 1 {\displaystyle {\begin{array}{rrcrcr}J_{A}=&-1&:&1&:&1\\J_{B}=&1&:&-1&:&1\\J_{C}=&1&:&1&:&-1\end{array}}} The radii of

3570-588: The midpoints of the sides). The radius of the incircle is related to the area of the triangle. The ratio of the area of the incircle to the area of the triangle is less than or equal to π / 3 3 {\displaystyle \pi {\big /}3{\sqrt {3}}} , with equality holding only for equilateral triangles . Suppose △ A B C {\displaystyle \triangle ABC} has an incircle with radius r {\displaystyle r} and center I {\displaystyle I} . Let

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3640-413: The plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations . The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. The maximal diameter (which corresponds to the long diagonal of

3710-506: The plane by translation. In three dimensions, hexagonal prisms with parallel opposite faces are called parallelohedrons and these can tessellate 3-space by translation. In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane. Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon

3780-403: The plane of a regular hexagon with circumradius R {\displaystyle R} , whose distances to the centroid of the regular hexagon and its six vertices are L {\displaystyle L} and d i {\displaystyle d_{i}} respectively, we have If d i {\displaystyle d_{i}} are the distances from

3850-517: The regular hexagonal tiling , {6,3}, with three hexagonal faces around each vertex. A regular hexagon can also be created as a truncated equilateral triangle , with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D 3 symmetry. A truncated hexagon, t{6}, is a dodecagon , {12}, alternating two types (colors) of edges. An alternated hexagon, h{6}, is an equilateral triangle , {3}. A regular hexagon can be stellated with equilateral triangles on its edges, creating

3920-540: The same plane. The interior of such a hexagon is not generally defined. A skew zig-zag hexagon has vertices alternating between two parallel planes. A regular skew hexagon is vertex-transitive with equal edge lengths. In three dimensions it will be a zig-zag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D 3d , [2 ,6] symmetry, order 12. The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons. The regular skew hexagon

3990-1441: The second expression is obtained the same way. The distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation I A ¯ ⋅ I A ¯ C A ¯ ⋅ A B ¯ + I B ¯ ⋅ I B ¯ A B ¯ ⋅ B C ¯ + I C ¯ ⋅ I C ¯ B C ¯ ⋅ C A ¯ = 1. {\displaystyle {\frac {{\overline {IA}}\cdot {\overline {IA}}}{{\overline {CA}}\cdot {\overline {AB}}}}+{\frac {{\overline {IB}}\cdot {\overline {IB}}}{{\overline {AB}}\cdot {\overline {BC}}}}+{\frac {{\overline {IC}}\cdot {\overline {IC}}}{{\overline {BC}}\cdot {\overline {CA}}}}=1.} Additionally, I A ¯ ⋅ I B ¯ ⋅ I C ¯ = 4 R r 2 , {\displaystyle {\overline {IA}}\cdot {\overline {IB}}\cdot {\overline {IC}}=4Rr^{2},} where R {\displaystyle R} and r {\displaystyle r} are

4060-431: The sides equals the radius of the circumscribed circle or circumcircle , which equals 2 3 {\displaystyle {\tfrac {2}{\sqrt {3}}}} times the apothem (radius of the inscribed circle ). All internal angles are 120 degrees . A regular hexagon has six rotational symmetries ( rotational symmetry of order six ) and six reflection symmetries ( six lines of symmetry ), making up

4130-512: The sides, incircle radius, and circumcircle radius are: a b + b c + c a = s 2 + ( 4 R + r ) r , a 2 + b 2 + c 2 = 2 s 2 − 2 ( 4 R + r ) r . {\displaystyle {\begin{aligned}ab+bc+ca&=s^{2}+(4R+r)r,\\a^{2}+b^{2}+c^{2}&=2s^{2}-2(4R+r)r.\end{aligned}}} Any line through

4200-401: The successive sides of a cyclic hexagon are a , b , c , d , e , f , then the three main diagonals intersect in a single point if and only if ace = bdf . If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are concurrent . If

4270-507: The supports there is a nine-meter vestibule with a ribbed arch extending nine meters. It seems that the stone-supports of the bridge were constructed during the reign of the Sassanids . Hexagonal In geometry , a hexagon (from Greek ἕξ , hex , meaning "six", and γωνία , gonía , meaning "corner, angle") is a six-sided polygon . The total of the internal angles of any simple (non-self-intersecting) hexagon

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4340-415: The term "hexagon" has prevailed in common usage and academic literature, solidifying its place in mathematical terminology despite the historical argument for "sexagon." The consensus remains that "hexagon" is the appropriate term, reflecting its Greek origins and established usage in mathematics. (see Numeral_prefix#Occurrences ). Inradius In geometry , the incircle or inscribed circle of

4410-455: The total area is: Δ = r 2 ( cot ⁡ A 2 + cot ⁡ B 2 + cot ⁡ C 2 ) . {\displaystyle \Delta =r^{2}\left(\cot {\tfrac {A}{2}}+\cot {\tfrac {B}{2}}+\cot {\tfrac {C}{2}}\right).} The Gergonne triangle (of △ A B C {\displaystyle \triangle ABC} )

4480-633: The touchpoints where the incircle touches B C ¯ {\displaystyle {\overline {BC}}} , A C ¯ {\displaystyle {\overline {AC}}} , and A B ¯ {\displaystyle {\overline {AB}}} . The incenter is the point where the internal angle bisectors of ∠ A B C , ∠ B C A ,  and  ∠ B A C {\displaystyle \angle ABC,\angle BCA,{\text{ and }}\angle BAC} meet. The trilinear coordinates for

4550-774: The triangle vertex positions. Barycentric coordinates for the incenter are given by   a : b : c {\displaystyle \ a:b:c} where a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are the lengths of the sides of the triangle, or equivalently (using the law of sines ) by sin ⁡ A : sin ⁡ B : sin ⁡ C {\displaystyle \sin A:\sin B:\sin C} where A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} are

4620-516: The triangle's circumradius and inradius respectively. The collection of triangle centers may be given the structure of a group under coordinate-wise multiplication of trilinear coordinates; in this group, the incenter forms the identity element . The distances from a vertex to the two nearest touchpoints are equal; for example: d ( A , T B ) = d ( A , T C ) = 1 2 ( b + c −

4690-426: The triangle's sides. The center of an excircle is the intersection of the internal bisector of one angle (at vertex A {\displaystyle A} , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex A {\displaystyle A} , or the excenter of A {\displaystyle A} . Because

4760-399: The triangle's sides. The center of the incircle, called the incenter , can be found as the intersection of the three internal angle bisectors . The center of an excircle is the intersection of the internal bisector of one angle (at vertex A , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex A , or

4830-422: The vertices of a regular hexagon to any point on its circumcircle, then The regular hexagon has D 6 symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D 6 ), 2 dihedral: (D 3, D 2 ), 4 cyclic : (Z 6 , Z 3 , Z 2 , Z 1 ) and the trivial (e) These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order. r12

4900-947: The vertices of the intouch triangle are given by T A = 0 : sec 2 ⁡ B 2 : sec 2 ⁡ C 2 T B = sec 2 ⁡ A 2 : 0 : sec 2 ⁡ C 2 T C = sec 2 ⁡ A 2 : sec 2 ⁡ B 2 : 0. {\displaystyle {\begin{array}{ccccccc}T_{A}&=&0&:&\sec ^{2}{\frac {B}{2}}&:&\sec ^{2}{\frac {C}{2}}\\[2pt]T_{B}&=&\sec ^{2}{\frac {A}{2}}&:&0&:&\sec ^{2}{\frac {C}{2}}\\[2pt]T_{C}&=&\sec ^{2}{\frac {A}{2}}&:&\sec ^{2}{\frac {B}{2}}&:&0.\end{array}}} Trilinear coordinates for

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