Let chords AC and BD of a circle ω intersect at P. A smaller circle ω1 is tangent to ω at T and to segments AP and DP at E and F respectively.
(a) Prove that ray T E bisects arc ABC of ω.
(b) Let I be the incenter of triangle ACD and M be the midpoint of arc ABC of ω. Prove that MA = MI = MC.
(c) Let F* be the common point of ω1 and line EI other than E. Prove that I, F0 , D, T are concyclic.
(d) Prove that DF* is tangent to ω1. This means that F = F* , so that E, F, I are collinear
I have proved a, b and c so may assume that these are true. I need help in proving that d is true.