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Prove that JM, KN and LP are concurrent

Source: Romanian TST 2011

April 9, 2012
geometryincentersearchgeometry proposed

Problem Statement

The incircle of a triangle ABCABC touches the sides BC,CA,ABBC,CA,AB at points D,E,FD,E,F, respectively. Let XX be a point on the incircle, different from the points D,E,FD,E,F. The lines XDXD and EF,XEEF,XE and FD,XFFD,XF and DEDE meet at points J,K,LJ,K,L, respectively. Let further M,N,PM,N,P be points on the sides BC,CA,ABBC,CA,AB, respectively, such that the lines AM,BN,CPAM,BN,CP are concurrent. Prove that the lines JM,KNJM,KN and LPLP are concurrent.
Dinu Serbanescu
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