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perpendicular bisector of BC bisects segment AD

Source: 2022 NZMO - New Zealand Maths Olympiad Round 1 p7

October 9, 2022
geometrybisects segment

Problem Statement

Let MM be the midpoint of side BCBC of acute triangle ABCABC. The circle centered at MM passing through AA intersects the lines ABAB and ACAC again at PP and QQ, respectively. The tangents to this circle at PP and QQ meet at DD. Prove that the perpendicular bisector of BCBC bisects segment ADAD.
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