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Finding a cyclic quadrilateral

Source: Indonesian MO (INAMO) 2009, Day 2, Problem 8

August 8, 2009
geometrytrigonometrycircumcircleangle bisectorperpendicular bisectorgeometry unsolved

Problem Statement

Given an acute triangle ABC ABC. The incircle of triangle ABC ABC touches BC,CA,AB BC,CA,AB respectively at D,E,F D,E,F. The angle bisector of ∠A \angle A cuts DE DE and DF DF respectively at K K and L L. Suppose AA1 AA_1 is one of the altitudes of triangle ABC ABC, and M M be the midpoint of BC BC. (a) Prove that BK BK and CL CL are perpendicular with the angle bisector of ∠BAC \angle BAC. (b) Show that A1KML A_1KML is a cyclic quadrilateral.
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