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182

Part of 1973 All Soviet Union Mathematical Olympiad

Problems(1)

ASU 182 All Soviet Union MO 1973 similar triangles on sides of triangle

Source:

7/4/2019
Three similar acute-angled triangles AC1B,BA1CAC_1B, BA_1C and CB1ACB_1A are constructed on the outer side of the acute-angled triangle ABCABC. (Equal triples of the angles are AB1C,ABC1,A1BCAB_1C, ABC_1, A_1BC and BA1C,BAC1,B1ACBA_1C, BAC_1, B_1AC.)
a) Prove that the circles circumscribed around the outer triangles intersect in one point.
b) Prove that the straight lines AA1,BB1AA_1, BB_1 and CC1CC_1 intersect in the same point
geometrysimilar trianglesconcurrent