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 be a point in the plane of the reference triangle. Let the lines intersect the circumcircle of at. The triangle is called the circumcevian triangle of with reference to.

Coordinates

Let be the side lengths of triangle and let the trilinear coordinates of be. Then the trilinear coordinates of the vertices of the circumcevian triangle of are as follows:

Some properties

Every triangle inscribed in the circumcircle of the reference triangle ABC is congruent to exactly one circumcevian triangle.
The circumcevian triangle of P is similar to the pedal triangle of P.
The McCay cubic is the locus of point P such that the circumcevian triangle of P and ABC are orthologic.