In a triangle ABC (see the figure below)
the incircle I is tangent to BC, AC, and AB at T1, T2, and T3.
The excircle E corresponding to BC is tangent to AC, BC, and AB
at F, G, and H, respectively. D is a point on AC so that the
incircles of triangles ABD and BDC are congruent. Prove that BD
is the geometric mean of AF and CF, that is,