The attached figure was constructed as follows:

Given Triangle ABC extend AC to E such that AE = AE1. AD1 is the angle bisector of Angle BAE and AD is the angle bisector of Angle BAC. O is the midpoint of DD1. AD//EB and E1B//AD1. Arc DAD1 is drawn.

I know that the following ratios exist:
AC/AE = DC/DB
AC/AE1 = D1C/D1B
DC/DB = D1C/D1B

Let CO = u, BO = v, and (1/2)DD1 = R. Prove that u*v = R^2

Circle of Apollonius-ratio.jpg