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

 

Circumcevian triangle 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 triangle ABC at A', B', C'. The triangle A'B'C' is called the circumcevian triangle of P with reference to the triangle 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] $${\displaystyle {\begin{array}{rccccc}A'=&-a\beta \gamma &:&(b\gamma +c\beta )\beta &:&(b\gamma +c\beta )\gamma \\B'=&(c\alpha +a\gamma )\alpha &:&-b\gamma \alpha &:&(c\alpha +a\gamma )\gamma \\C'=&(a\beta +b\alpha )\alpha &:&(a\beta +b\alpha )\beta &:&-c\alpha \beta \end{array}}}$$ 

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

References

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