 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

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 "ΔABC", 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.