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0361 geometry 3rd edition Round 6 p1

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May 9, 2021
geometry3rd edition

Problem Statement

For a triangle ABCABC and a point MM inside the triangle we consider the lines AM,BM,CMAM, BM,CM which intersect the sides BC,CA,ABBC, CA, AB in A1,B1,C1A_1, B_1, C_1 respectively. Take A′,B′,C′A', B', C' to be the intersection points between the lines AA1,BB1,CC1AA_1, BB_1, CC_1 and B1C1,C1A1,A1B1B_1C_1, C_1A_1, A_1B_1 respectively. a) Prove that the lines BC′,CB′BC', CB' and AA′AA' intersect in a point A2A_2; b) Define similarly points B2,C2B_2, C_2. Find the loci of MM such that the triangle A1B1C1A_1B_1C_1 is similar with the triangle A2B2C2A_2B_2C_2.
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