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.

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

Some properties

• Every triangle inscribed in the circumcircle of the reference triangle ABC is congruent to exactly one circumcevian triangle.
• 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