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4 semicircles inside a right triangle

Source: JBMO Shortlist 2004

October 13, 2017
JBMOgeometry

Problem Statement

Let ABCABC be a triangle with m(∠C)=90∘m (\angle C) = 90^\circ and the points D∈[AC],E∈[BC]D \in [AC], E\in [BC]. Inside the triangle we construct the semicircles C1,C2,C3,C4C_1, C_2, C_3, C_4 of diameters [AC],[BC],[CD],[CE][AC], [BC], [CD], [CE] and let {C,K}=C1∩C2,{C,M}=C3∩C4,{C,L}=C2∩C3,{C,N}=C1∩C4\{C, K\} = C_1 \cap C_2, \{C, M\} =C_3 \cap C_4, \{C, L\} = C_2 \cap C_3, \{C, N\} =C_1 \cap C_4. Show that points K,L,M,NK, L, M, N are concyclic.
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