Given three points A, B and C,
find the point P such that angle(PAB) = angle(PBC) = angle(PCA). Express tan(PAB) in terms of A, B and C. 

Constructions for a geometric proof. Not easy, it took me a few days, and I eventually had to back it out of an analytic calculation.
(applet)tan(PAB) = tan(B1,A1,B2) = (B1,B2)/(A1,B1) = AB/((A1,A) + (A,B1)) using angle(A1,A,C) = right_angle  angle(A,A1,C) angle(A,A1,C) = angle(CAB) III 32 tan(PAB) = AB/(AC/sin(CAB) + AB/tan(ABC)) simplifying notation, tan(PAB) = c/(b/sin(A) + c/tan(B)) and the expressions c/(b/sin(A) + c/tan(B)), a/(c/sin(B) + a/tan(C)), b/(a/sin(C) + b/tan(A)) must all be equal.