IMO - SL-2008 - G4

In an acute triangle ABC segments BE and CF are altitudes. Two circles passing through the point A anf F and tangent to the line BC at the points P and Q so that B lies between C and Q. Prove that lines PE and QF intersect on the circumcircle of triangle AEF.

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3rd Iranian Geometry Olympiad-P4

Let ω be the circumcircle of right-angled triangle ABC (∠A = 90◦). Tangent to ω at point A intersects the line BC in point P. Suppose that M is the midpoint of (the smaller) arc AB, and PM intersects ω for the second time in Q. Tangent to ω at point Q intersects AC in K. Prove that ∠PKC = 90◦.

Proposed by Davood Vakili.