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<MKN=90^o wanted, midpoint and 2 arc midpoints, intersecting circles

Source: Indonesia INAMO Shortlist 2010 G9

August 27, 2021
right anglearc midpointcirclesgeometry

Problem Statement

Given two circles Γ1\Gamma_1 and Γ2\Gamma_2 which intersect at points AA and BB. A line through AA intersects Γ1\Gamma_1 and Γ2\Gamma_2 at points CC and DD, respectively. Let MM be the midpoint of arc BCBC in Γ1\Gamma_1 ,which does not contains AA, and NN is the midpoint of the arc BDBD in Γ2\Gamma_2, which does not contain AA. If KK is the midpoint of CDCD, prove that ∠MKN=90o.\angle MKN = 90^o.
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