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

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Let Template:Mvar be a point in the plane of the reference triangle Template:Math. Let the lines Template:Mvar intersect the circumcircle of Template:Math at Template:Mvar. The triangle Template:Math is called the circumcevian triangle of Template:Mvar with reference to Template:Math.

Let Template:Mvar be the side lengths of triangle Template:Math and let the trilinear coordinates of Template:Mvar be Template:Math. Then the trilinear coordinates of the vertices of the circumcevian triangle of Template:Mvar are as follows: $$\begin{array}{cccccc}{A}^{\prime }=& -a\beta \gamma & :& (b\gamma +c\beta )\beta & :& (b\gamma +c\beta )\gamma \\ B^{\prime }=& (c\alpha +a\gamma )\alpha & :& -b\gamma \alpha & :& (c\alpha +a\gamma )\gamma \\ C^{\prime }=& (a\beta +b\alpha )\alpha & :& (a\beta +b\alpha )\beta & :& -c\alpha \beta \end{array}$$

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