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
[[File:CircumCevianTriangle.png|thumb|

{{legend|red|Point $P$}}

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Let $P$ be a point in the plane of the reference triangle $△ABC$. Let the lines $P$ intersect the circumcircle of $△ABC$ at $P$. The triangle $△ABC$ is called the circumcevian triangle of $AP, BP, CP$ with reference to $△A'B'C'$.

Coordinates
Let $A', B', C'$ be the side lengths of triangle $△ABC$ and let the trilinear coordinates of $P$ be $△ABC$. Then the trilinear coordinates of the vertices of the circumcevian triangle of $a,b,c$ 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.