[math-fun] cyclic quadrilaterals

Given a generic triangle ABC, in "how many ways" can you choose an interior point P and points D, E, and F on sides BC, AC, and AB (respectively) so that quadrilaterals AEPF, BFPD, and CDPE are cyclic? That is to say, how many degrees of freedom do you have? Is there a nice way to parametrize the set of such configurations (with A,B,C fixed), so as to express the lengths of segments AP, BP, CP, DP, EP, and FP in terms of the lengths of segments AB, AC, and BC together with the extra parameter(s)? B / \ / \ F D / P \ / \ A-----E-----C Jim Propp University of Wisconsin Madison, WI