The Conjugate Hyperbola and the Involutory Ellipse

Extract

N.B.—In what follows we consider the conic as the projection of the circle and assume only its descriptive (including cross-ratio) properties as known.

1. A circle, centre g₁ is cut by a secant line at ww′. Any line cpp′ through c, the pole of ww′, cuts the circle at p and again at p′. The lines wp and w′p′ intersect at d. The locus of d for the variable secant line through c is a hyperbola touching the given circle at ww′ (If w′p and wp′ meet at d′, the locus of d′ is the same hyperbola.)