Circle and equal angles.

[ins-]

The circle k is tangent to the sides AD, BC and CD of the quadrilateral ABCD at the points A, B and E - the tangent point of k with the side CD. F is the foot of perpendicular from E to AB. Prove that <CFE=<DFE.

[Ilya Nek]

Olimpiad problem? Triangle $BCF$ similar to triangle $FAD$.

[ins-]

Why they are similar?

[Ilya Nek]

\angle$FBC$=\angle$FAD$ and $BF$/$BC$=$AF$/$AD$ (cosines help, because \angle$FBE$=\angle$AED$, \angle$FAE$=\angle$EBC$)

[ins-]

I got it. Thank you! It is definitely not a hard problem if you see the facts mentioned. I think it is a good problem for earlier stages of some math olympiad.

[TOTAL]

$$\tg z = \frac{CE \cos x}{CB \cos y}= \frac{\cos x}{\cos y}$$

[ins-]

Sometimes a picture says more than a thousands words. Thank you for the valuable and beautiful idea.