MathDB
Hard geo

Source: VMEO III

March 30, 2016
geometry

Problem Statement

Given a triangle ABCABC, incircle (I)(I) touches BC,CA,ABBC,CA,AB at D,E,FD,E,F respectively. Let MM be a point inside ABCABC. Prove that MM lie on (I)(I) if and only if one number among AESBMC,BFSCMA,CDSAMB\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}} is sum of two remaining numbers (SABCS_{ABC} denotes the area of triangle ABCABC)