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collinear, perpendicular and midpoint wanted, 2 circles related

Source: 2016 Saudi Arabia GMO TST level 4+, I p3

August 1, 2020
geometrycollinearmidpointperpendicular bisector

Problem Statement

Let ABCABC be an acute, non-isosceles triangle with the circumcircle (O)(O). Denote D,ED, E as the midpoints of AB,ACAB,AC respectively. Two circles (ABE)(ABE) and (ACD)(ACD) intersect at KK differs from AA. Suppose that the ray AKAK intersects (O)(O) at LL. The line LBLB meets (ABE)(ABE) at the second point MM and the line LCLC meets (ACD)(ACD) at the second point NN. a) Prove that M,K,NM, K, N collinear and MNMN perpendicular to OLOL. b) Prove that KK is the midpoint of MNMN
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