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ABC necessarily isosceles ? All-Russian MO 1993 Regional (R4) 9.3

Source:

August 26, 2024
geometryisosceles

Problem Statement

Points MM and NN are chosen on the sides ABAB and BC of a triangle ABCABC. The segments ANAN and CMCM meet at OO such that AO=COAO =CO. Is the triangle ABCABC necessarily isosceles, if
(a) AM=CNAM = CN?
(b) BM=BNBM = BN?
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