Circumcevian triangle
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.
Definition
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
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] [math]\displaystyle{ \begin{array}{rccccc} A' =& -a\beta\gamma &:& (b\gamma+c\beta)\beta &:& (b\gamma+c\beta)\gamma \\ B' =& (c\alpha +a\gamma)\alpha &:& - b\gamma\alpha &:& (c\alpha +a\gamma) \gamma \\ C' =& (a\beta +b\alpha)\alpha &:& (a\beta +b\alpha)\beta &:& - c\alpha\beta \end{array} }[/math]
Some properties
- 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]
See also
References
- ↑ Kimberling, C (1998). "Triangle Centers and Central Triangles". Congress Numerantium 129: 201.
- ↑ 2.0 2.1 2.2 Weisstein, Eric W.. ""Circumcevian Triangle"". MathWorld. https://mathworld.wolfram.com/CircumcevianTriangle.html.
- ↑ Bernard Gilbert. "K003 McCay Cubic". Bernard Gilbert. https://bernard-gibert.pagesperso-orange.fr/Exemples/k003.html.
Original source: https://en.wikipedia.org/wiki/Circumcevian triangle.
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