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 the sidelengths of △ABC . 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 △ABC 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 △ABC and is given by the equation

Centers and tangent lines

Circumconic

The center of the general circumconic is the point The lines tangent to the general circumconic at the vertices A, B, C are, respectively,

Inconic

The center of the general inconic is the point The lines tangent to the general inconic are the sidelines of △ABC , given by the equations x = 0 , y = 0 , z = 0 .

Other features

Circumconic

△ABC in a point other than A, B, C, often called the fourth point of intersection, given by trilinear coordinates

Inconic

X2 is the inconic, necessarily an ellipse, given by the equation (α, β, γ) of the inellipse's center, is α = β = γ = ⅓ .

Extension to quadrilaterals

All the centers of inellipses of a given quadrilateral fall on the line segment connecting the midpoints of the diagonals of the quadrilateral.

Examples