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.

See also

• Cevian
• Ceva's theorem

References