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Points on the sides of triangle

Source: VMO 2019

April 15, 2019
geometryincenterparallelogramorthocenterarea of a triangle

Problem Statement

Let ABCABC be triangle with HH is the orthocenter and II is incenter. Denote A1,A2,B1,B2,C1,C2A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2} be the points on the rays AB,AC,BC,CA,CBAB, AC, BC, CA, CB, respectively such that AA1=AA2=BC,BB1=BB2=CA,CC1=CC2=AB.AA_{1} = AA_{2} = BC, BB_{1} = BB_{2} = CA, CC_{1} = CC_{2} = AB. Suppose that B1B2B_{1}B_{2} cuts C1C2C_{1}C_{2} at A′A', C1C2C_{1}C_{2} cuts A1A2A_{1}A_{2} at B′B' and A1A2A_{1}A_{2} cuts B1B2B_{1}B_{2} at C′C'. a) Prove that area of triangle A′B′C′A'B'C' is smaller than or equal to the area of triangle ABCABC. b) Let JJ be circumcenter of triangle A′B′C′A'B'C'. AJAJ cuts BCBC at RR, BJBJ cuts CACA at SS and CJCJ cuts ABAB at TT. Suppose that (AST),(BTR),(CRS)(AST), (BTR), (CRS) intersect at KK. Prove that if triangle ABCABC is not isosceles then HIJKHIJK is a parallelogram.
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