The Theory of the General Contact Circles of a Triangle

With every point P in the plane of a triangle may be associated two circles-its pedal and contact circles. The former is the circle through the projections of P on the sides of the triangle, while the latter is the circle through the points of contact with the sides, of the unique inconic of the triangle whose centre is P. It is known that a pair of isogonal conjugates P, P' w. r. t. a triangle determine a common pedal circle and a common contact circle. Now, pairs of isogonal conjugates w. r. t. a triangle are but the singular members of a linear ∞³-system of conic envelopes-called infocal conics (3.1)-inpolar to the rectangular hyperbolas through the in and ex-centres of the triangle. This suggests the possibility of associating with every infocal conic a pedal circle and a contact circle.