Consider the quadrilateral [math]PQRP_1Q_1R_1[/math] with [math]P[/math] conjugate to [math]P_1[/math] and [math]Q[/math] to [math]Q_1[/math]. The polars [math]p[/math] and [math]q[/math] pass through [math]P_1[/math] and [math]Q_1[/math], respectively. By Theorem 7.31, the polar of [math]R[/math] meets [math]PQ[/math] in a point that lies on [math]P_1Q_1[/math], namely in the point [math]PQ\bullet P_1Q_1 = R_1[/math]. Therefore the polar of [math]R[/math] passes through [math]R_1[/math] so they are conjugates.