Hyperbola Proof

Since I've answered a few questions on here from time to time, now, I've got one!!

Draw a tangent line to either the right (or left) branch of a hyperbola and let that line intersect the x axis. Call this point of intersection, "S." From the point of tangency - call it point A - draw two lines, one to each of the focal points of the hyperbola. Call these points (-C) and (C). So we have a "big" triangle, -CAC, and two smaller trangles, -CAS and CAS.

According to a theorem, < -CAS = < CAS.

Does anyone know how to prove this??

(For convenience, you can let the hyperbola be centered at (0,0) so that points -C and C lie on the x axis.)

I've tried proving it by noting that since, in the "big" triangle -CAC the apex angle is bisected, then by Euclid VI 3, -CS * AC = CS *-CA.

Also, sin(-CAS) = sin(CAS) since they're equal angles. And sin(-CSA) = sin(CSA) since they're supplementary. Then, this implies that sin(-CAS) * sin(CSA) = sin(CAS) * sin(-CSA).

My "proof" almost "works" except for one small problem..........one side of the proof is the negative of the other!!

One hint that might help: Set -CS, CS and AS up as "distances," and use Euclid's Theorem. (This is only a suggestion. You may approach it in any manner you'd like!!)

Anyone have any ideas??