In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle,[1] and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.
[2] 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.
In trilinear coordinates, the general circumconic is the locus of a variable point
The general inconic is tangent to the three sidelines of △ABC and is given by the equation The center of the general circumconic is the point The lines tangent to the general circumconic at the vertices A, B, C are, respectively, 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.
All the centers of inellipses of a given quadrilateral fall on the line segment connecting the midpoints of the diagonals of the quadrilateral.