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JBMO Shortlist 2019 G7

Source:

September 12, 2020
geometry

Problem Statement

Let ABCABC be a right-angled triangle with ∠A=90∘\angle A = 90^{\circ}. Let KK be the midpoint of BCBC, and let AKLMAKLM be a parallelogram with centre CC. Let TT be the intersection of the line ACAC and the perpendicular bisector of BMBM. Let ω1\omega_1 be the circle with centre CC and radius CACA and let ω2\omega_2 be the circle with centre TT and radius TBTB. Prove that one of the points of intersection of ω1\omega_1 and ω2\omega_2 is on the line LMLM.
Proposed by Greece
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