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perpendiculars on a convex ABCD with 2 angles 135^o

Source: Czech-Polish-Slovak Match 2014 day 1 P3

September 29, 2017
perpendiculargeometryconvex quadrilateral

Problem Statement

Given is a convex ABCDABCD, which is ∣∠ABC∣=∣∠ADC∣=135∘ |\angle ABC| = |\angle ADC|= 135^\circ . On the AB,ADAB, AD are also selected points M,NM, N such that ∣∠MCD∣=∣∠NCB∣=90∘ |\angle MCD| = |\angle NCB| = 90^ \circ . The circumcircles of the triangles AMNAMN and ABDABD intersect for the second time at point K≠AK \ne A. Prove that lines AKAK and KCKC are perpendicular.
(Irán)
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