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2018 VNTST Problem 1

Source: 2018 Vietnam Team Selection Test

March 30, 2018
geometrycircumcircleEuler

Problem Statement

Let ABCABC be a acute, non-isosceles triangle. D, E, FD,\ E,\ F are the midpoints of sides AB, BC, ACAB,\ BC,\ AC, resp. Denote by (O), (O′)(O),\ (O') the circumcircle and Euler circle of ABCABC. An arbitrary point PP lies inside triangle DEFDEF and DP, EP, FPDP,\ EP,\ FP intersect (O′)(O') at D′, E′, F′D',\ E',\ F', resp. Point A′A' is the point such that D′D' is the midpoint of AA′AA'. Points B′, C′B',\ C' are defined similarly. a. Prove that if PO=PO′PO=PO' then O∈(A′B′C′)O\in(A'B'C'); b. Point A′A' is mirrored by ODOD, its image is XX. Y, ZY,\ Z are created in the same manner. HH is the orthocenter of ABCABC and XH, YH, ZHXH,\ YH,\ ZH intersect BC,AC,ABBC, AC, AB at M, N, LM,\ N,\ L resp. Prove that M, N, LM,\ N,\ L are collinear.
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