P, Q and R are three points located on a circle L with diameter 4 and satisfying PQ = QR. Point S is located inside L in such a manner that QR = RS = SQ. The line passing through P and S intersects L at the point T.

Determine the length of ST.

I can find an S very simply but the part I'm still working on is proving that my S is unique.  My gut reaction upon first reading the problem was that S was actually the center.  So I tried seeing if a valid P,Q,R could be choosen starting with S the center of a circle with radius 2.  Chosing an arbitrary Q on the circle R and T are respectivly determined the fact that triangles QSR and QST are both equilateral.  As it turns out PR ends up being 2*Sqrt(3).  Using this S we have that ST is a radius of the circle and thus  ST=2.  I leave the proof of uniqueness to someone of greater geometric skill than myself.

Posted by Daniel on 2008-01-25 11:43:33