Circumconic and inconic

In triangle geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.

Suppose A,B,C are distinct non-collinear points, and let ΔABC denote the triangle whose vertices are A,B,C. Following common practice, A denotes not only the vertex but also the angle BAC at vertex A, and similarly for B and C as angles in ΔABC. Let a = |BC|, b = |CA|, c = |AB|, the sidelengths of ΔABC.

In trilinear coordinates, the general circumconic is the locus of a variable point X = x : y : z satisfying an equation

for some point u : v : w. The isogonal conjugate of each point X on the circumconic, other than A,B,C, is a point on the line

This line meets the circumcircle of ΔABC in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola.

The general inconic is tangent to the three sidelines of ΔABC and is given by the equation

Centers and tangent lines