Circumcevian triangle

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Short description: Triangle derived from a given triangle and a coplanar point

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

  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

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

  1. Kimberling, C (1998). "Triangle Centers and Central Triangles". Congress Numerantium 129: 201. 
  2. 2.0 2.1 2.2 Weisstein, Eric W.. ""Circumcevian Triangle"". MathWorld. https://mathworld.wolfram.com/CircumcevianTriangle.html. 
  3. Bernard Gilbert. "K003 McCay Cubic". Bernard Gilbert. https://bernard-gibert.pagesperso-orange.fr/Exemples/k003.html.