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show that quadrilateral is cyclic and its circumcircle contains the incenter

Source: Greece JBMO TST 2017, Problem 2

June 25, 2018
geometryGreece

Problem Statement

Let ABCABC be an acute-angled triangle inscribed in a circle C(O,R)\mathcal C (O, R) and FF a point on the side ABAB such that AF<AB/2AF < AB/2. The circle c1(F,FA)c_1(F, FA) intersects the line OAOA at the point A′A' and the circle C\mathcal C at KK. Prove that the quadrilateral BKFA′BKFA' is cyclic and its circumcircle contains point OO.
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