In Euclidean geometry, a circumcevian triangle is a special triangle associated with a reference triangle and a point in the plane of the triangle. It is also associated with the circumcircle of the reference triangle.
Circumcircle of △ABC; lines between the vertices of △ABC and P
Circumcevian triangle △A'B'C' of P
Let P be a point in the plane of the reference triangle △ABC. Let the lines AP, BP, CP intersect the circumcircle of △ABC at A', B', C'. The triangle △A'B'C' is called the circumcevian triangle of P with reference to △ABC.[1]
Coordinatesedit
Let a,b,c be the side lengths of triangle △ABC and let the trilinear coordinates of P be α : β : γ. Then the trilinear coordinates of the vertices of the circumcevian triangle of P are as follows:[2]
Some propertiesedit
Every triangle inscribed in the circumcircle of the reference triangle ABC is congruent to exactly one circumcevian triangle.[2]
The circumcevian triangle of P is similar to the pedal triangle of P.[2]
The McCay cubic is the locus of point P such that the circumcevian triangle of P and ABC are orthologic.[3]
^Kimberling, C (1998). "Triangle Centers and Central Triangles". Congress Numerantium. 129: 201.
^ abcWeisstein, Eric W. ""Circumcevian Triangle"". From MathWorld--A Wolfram Web Resource. MathWorld. Retrieved 24 December 2021.
^Bernard Gilbert. "K003 McCay Cubic". Catalogue of Triangle Cubics. Bernard Gilbert. Retrieved 24 December 2021.
January 01, 1970
circumcevian, triangle, euclidean, geometry, circumcevian, triangle, special, triangle, associated, with, reference, triangle, point, plane, triangle, also, associated, with, circumcircle, reference, triangle, contents, definition, coordinates, some, propertie. In Euclidean geometry a circumcevian triangle is a special triangle associated with a reference triangle and a point in the plane of the triangle It is also associated with the circumcircle of the reference triangle Contents 1 Definition 2 Coordinates 3 Some properties 4 See also 5 ReferencesDefinition edit nbsp Reference triangle ABC Point P Circumcircle of ABC lines between the vertices of ABC and P Circumcevian triangle A B C of P Let P be a point in the plane of the reference triangle ABC Let the lines AP BP CP intersect the circumcircle of ABC at A B C The triangle A B C is called the circumcevian triangle of P with reference to ABC 1 Coordinates editLet a b c be the side lengths of triangle ABC and let the trilinear coordinates of P be a b g Then the trilinear coordinates of the vertices of the circumcevian triangle of P are as follows 2 A a b g b g c b b b g c b g B c a a g a b g a c a a g g C a b b a a a b b a b c a b displaystyle begin array rccccc A amp a beta gamma amp amp b gamma c beta beta amp amp b gamma c beta gamma B amp c alpha a gamma alpha amp amp b gamma alpha amp amp c alpha a gamma gamma C amp a beta b alpha alpha amp amp a beta b alpha beta amp amp c alpha beta end array nbsp Some properties editEvery triangle inscribed in the circumcircle of the reference triangle ABC is congruent to exactly one circumcevian triangle 2 The circumcevian triangle of P is similar to the pedal triangle of P 2 The McCay cubic is the locus of point P such that the circumcevian triangle of P and ABC are orthologic 3 See also editCevian Ceva s theoremReferences edit Kimberling C 1998 Triangle Centers and Central Triangles Congress Numerantium 129 201 a b c Weisstein Eric W Circumcevian Triangle From MathWorld A Wolfram Web Resource MathWorld Retrieved 24 December 2021 Bernard Gilbert K003 McCay Cubic Catalogue of Triangle Cubics Bernard Gilbert Retrieved 24 December 2021 Retrieved from https en wikipedia org w index php title Circumcevian triangle amp oldid 1179356117, wikipedia, wiki, book, books, library,
