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triangle and orthocente

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September 28, 2017
geometry

Problem Statement

Consider the acute-angled triangle ABCABC. Let XX be a point on the side BCBC, and YY be a point on the side CACA. The circle k1k_1 with diameter AXAX cuts ACAC again at E′E' .The circle k2k_2 with diameter BYBY cuts BCBC again at B′B'. (i) Let MM be the midpoint of XYXY . Prove that A′M=B′MA'M = B'M. (ii) Suppose that k1k_1 and k2k_2 meet at PP and QQ. Prove that the orthocentre of ABCABC lies on the line PQPQ.
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