Semicircle and middle

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The segment AB is a diameter of semicircle k. From A and B are drawn the tangents to k - ta and tb. On k is chosen a random point C different from A and B. The tangent to k from C intersects ta and tb at the points D and E respectively. F is the intersection point of AE and BC. The line DF intersects BE at point P. Prove that P is the middle of BE.

The problem is probably well known it remains me: http://www.math.ca/Competitions/CMO/examt/english74.pdf - problem 5.

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http://www.artofproblemsolving.com/Foru ... 8&t=366455
http://www.math10.com/f/viewtopic.php?f=10&t=3096l

One solution on two sites...

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