Circumcevian triangle explained

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 .^[1]

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:^[2]

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

• Cevian
• Ceva's theorem

Notes and References

1. Kimberling, C . Triangle Centers and Central Triangles . Congress Numerantium . 1998 . 129 . 201.
2. Web site: Weisstein, Eric W. . "Circumcevian Triangle" . From MathWorld--A Wolfram Web Resource. . MathWorld . 24 December 2021.
3. Web site: Bernard Gilbert . K003 McCay Cubic . Catalogue of Triangle Cubics . Bernard Gilbert . 24 December 2021.