HMMT 二月 2022 · 几何 · 第 1 题

HMMT February 2022 — Geometry — Problem 1

Let ABC be a triangle with ∠ A = 60 . Line ℓ intersects segments AB and AC and splits triangle ABC into an equilateral triangle and a quadrilateral. Let X and Y be on ℓ such that lines BX and CY are perpendicular to ℓ . Given that AB = 20 and AC = 22, compute XY .

解析

Let ABC be a triangle with ∠ A = 60 . Line ℓ intersects segments AB and AC and splits triangle ABC into an equilateral triangle and a quadrilateral. Let X and Y be on ℓ such that lines BX and CY are perpendicular to ℓ . Given that AB = 20 and AC = 22, compute XY . Proposed by: Akash Das Answer: 21 ′ ′ ′ ′ Solution: Let the intersection points of ℓ with AB and AC be B and C . Note that AB + AC = ′ ′ ′ ′ ′ ′ 2 B C , BB = 2 XB , and CC = 2 Y C . Adding gives us ′ ′ ′ ′ ′ ′ ′ ′ AB + AC = AB + AC + BB + CC = 2( B C + XB + Y C ) = 2 XY. 20+22 Thus, XY = = 21. 2