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KM +ML = BC if AB = AC, <A = 30^o, AL = CM, <AMK = 45^o, <LMC = 75^ο

Source: 2023 IGO Elementary P2

January 18, 2024
geometryequal segmentsisosceles

Problem Statement

In an isosceles triangle ABCABC with AB=ACAB = AC and ∠A=30o\angle A = 30^o, points LL and MM lie on the sides ABAB and ACAC, respectively such that AL=CMAL = CM. Point KK lies on ABAB such that ∠AMK=45o\angle AMK = 45^o. If ∠LMC=75o\angle LMC = 75^o, prove that KM+ML=BCKM +ML = BC.
Proposed by Mahdi Etesamifard - Iran
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