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Something about Archimedes' lemma

Source: 2019 Belarus Team Selection Test 6.1

September 2, 2019
geometrygeometric transformationreflection

Problem Statement

Two circles Ω\Omega and Γ\Gamma are internally tangent at the point BB. The chord ACAC of Γ\Gamma is tangent to Ω\Omega at the point LL, and the segments ABAB and BCBC intersect Ω\Omega at the points MM and NN. Let M1M_1 and N1N_1 be the reflections of MM and NN about the line BLBL; and let M2M_2 and N2N_2 be the reflections of MM and NN about the line ACAC. The lines M1M2M_1M_2 and N1N2N_1N_2 intersect at the point KK. Prove that the lines BKBK and ACAC are perpendicular.
(M. Karpuk)
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