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The Only One Can Beat 9 Points, is 9-Point

Source: 2019 Taiwan TST Round 1

March 31, 2020
geometrygeometry proposed

Problem Statement

Given a triangle △ABC \triangle{ABC} with orthocenter H H . On its circumcenter, choose an arbitrary point P P (other than A,B,C A,B,C ) and let M M be the mid-point of HP HP . Now, we find three points D,E,F D,E,F on the line BC,CA,AB BC, CA, AB , respectively, such that AP∥HD,BP∥HE,CP∥HF AP \parallel HD, BP \parallel HE, CP \parallel HF . Show that D,E,F,M D, E, F, M are colinear.
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