John Conway writes:
Given P, there's a 1-parameter family. You can get them all by dropping "isoclines" from P to the sides - ie., lines such that angles AFP, BDP, CEP are all equal. Alternatively, let AEPF be any circle through A and P, and then BFPD and CEPD be the circumcircles of BFP and CEP (which will automatically intersect at a point DE on BC).
Thanks, John. Since this gives us 3 real degrees of freedom in total, I wonder if we get a nice parametrization of the full set of configurations (for A,B,C fixed) by specifying the three lengths AP, BP, and CP (or perhaps the three lengths DP, EP, and FP, or perhaps the lengths AD, BE, and CF, or ...). What I ideally want is a set of three parameters such that the distances between all the points can be expressed as rational functions of the parameters and the original lengths AB, AC, BC. Is this too much to hope for? Jim