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Source: VMO 2021 P3

December 25, 2020
geometrycircumcircle

Problem Statement

Let △ABC\bigtriangleup ABC is not an isosceles triangle and is an acute triangle, AD,BE,CFAD,BE,CF be the altitudes and HH is the orthocenter .Let II is the circumcenter of △HEF\bigtriangleup HEF and let K,JK,J is the midpoint of BC,EFBC,EF respectively.Let HJHJ intersects (I)(I) again at GG and GKGK intersects (I)(I) at L≠GL\neq G. a) Prove that ALAL is perpendicular to EFEF. b) Let ALAL intersects EFEF at MM, the line IMIM intersects the circumcircle △IEF\bigtriangleup IEF again at NN, DNDN intersects AB,ACAB,AC at PP and QQ respectively then prove that PE,QF,AKPE,QF,AK are concurrent.
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