Thanks, but this is only putting geometrically what I put algebraically, and doesn't really show what's going on, because your eventual equation is one between four-dimensional volumes. There ought to be a solution which stays in at most two dimensions. R.
there ought?? that the final relation
[BPQ]^2 = [ ... ] = ([ABQ] + [BCP] + [PDQ])^2 - 4 [ABQ] [BCP] .
is quadratic in areas suggests that there won't be such a solution. my solution had the mild benefit that being quadratic in areas arose from ratios of edge lengths. but yes, after that it's all algebraic. by the way, we went down a similar path some time ago when someone asked a related question about hero(n)'s formula: is there a reasonable 4-dimensional interpretation for s(s - a)(s - b)(s - c) ? i don't think anyone posted one. mike
B----F---------C /|\_ / / / / /_ / / | / \_ / / / / \_ / / | / \/ E----X------_--P / |/ __-- / / /_-- / A----Q---------D