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Is the tetrahedron regular?

Source: Austrian-Polish 2006 Question3

August 21, 2006
geometry3D geometrytetrahedrongeometry proposed

Problem Statement

ABCDABCD is a tetrahedron. Let KK be the center of the incircle of CBDCBD. Let MM be the center of the incircle of ABDABD. Let LL be the gravycenter of DACDAC. Let NN be the gravycenter of BACBAC. Suppose AKAK, BLBL, CMCM, DNDN have one common point. Is ABCDABCD necessarily regular?
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