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

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] ${\begin{array}{rccccc}A'=&-a\beta \gamma &:&(b\gamma +c\beta )\beta &:&(b\gamma +c\beta )\gamma \\B'=&(c\alpha +a\gamma )\alpha &:&-b\gamma \alpha &:&(c\alpha +a\gamma )\gamma \\C'=&(a\beta +b\alpha )\alpha &:&(a\beta +b\alpha )\beta &:&-c\alpha \beta \end{array}}$

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]

References

^ 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.