Circumconic and inconic

In Euclidean 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 △ABC 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 △ABC. Let ${\displaystyle a=|BC|,b=|CA|,c=|AB|,}$ the sidelengths of △ABC.

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

${\displaystyle uyz+vzx+wxy=0,}$

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

${\displaystyle ux+vy+wz=0.}$

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

In barycentric coordinates, the general inconic is tangent to the three sidelines of △ABC and is given by the equation

${\displaystyle u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz-2wuzx-2uvxy=0.}$