HMMT 二月 2022 · 团队赛 · 第 5 题

HMMT February 2022 — Team Round — Problem 5

[40] Let ABC be a triangle with centroid G , and let E and F be points on side BC such that BE = EF = F C . Points X and Y lie on lines AB and AC , respectively, so that X , Y , and G are not collinear. If the line through E parallel to XG and the line through F parallel to Y G intersect at P ̸ = G , prove that GP passes through the midpoint of XY . 4 3 2

解析

[40] Let ABC be a triangle with centroid G , and let E and F be points on side BC such that BE = EF = F C . Points X and Y lie on lines AB and AC , respectively, so that X , Y , and G are not collinear. If the line through E parallel to XG and the line through F parallel to Y G intersect at P ̸ = G , prove that GP passes through the midpoint of XY . Proposed by: Eric Shen CG Solution: Let CG intersect AB at N . Then N is the midpoint of AB and it is known that = 2 = AB CE , so EG ∥ AB . Moreover, since F E = EB , we have [ EF G ] = [ EXG ]. Similarly, [ EF G ] = [ F Y G ]. EB Now we have [ P XG ] = [ EXG ] = [ EF G ] = [ F Y G ] = [ P Y G ], so P G bisects XY , as desired. 4 3 2