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Some incircles...

Source: GMB-IMAR 2005, Seniors, Problem 1

October 9, 2005
geometryincenterrectanglecircumcircleperpendicular bisectorgeometry proposed

Problem Statement

The incircle of triangle ABCABC touches the sides BC,CA,ABBC,CA,AB at the points D,E,FD,E,F, respectively. Let KK be a point on the side BCBC and let MM be the point on the line segment AKAK such that AM=AE=AFAM=AE=AF. Denote by L,NL,N the incenters of triangles ABK,ACKABK,ACK, respectively. Prove that KK is the foot of the altitude from AA if and only if DLMNDLMN is a square. [hide="Remark"]This problem is slightly connected to [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=344774#p344774]GMB-IMAR 2005, Juniors, Problem 2 Bogdan Enescu
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