HMMT 二月 2003 · 冲刺赛 · 第 33 题

HMMT February 2003 — Guts Round — Problem 33

[10] We are given triangle ABC , with AB = 9, AC = 10, and BC = 12, and a point ′ ′ D on BC . B and C are reflected in AD to B and C , respectively. Suppose that lines ′ ′ BC and B C never meet (i.e., are parallel and distinct). Find BD . 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND

解析

We are given triangle ABC , with AB = 9, AC = 10, and BC = 12, and a point D on ′ ′ ′ BC . B and C are reflected in AD to B and C , respectively. Suppose that lines BC ′ and B C never meet (i.e., are parallel and distinct). Find BD . Solution: 6 The lengths of AB and AC are irrelevant. Because the figure is symmetric about AD , ′ ′ lines BC and B C meet if and only if they meet at a point on line AD . So, if they ′ never meet, they must be parallel to AD . Because AD and BC are parallel, triangles ′ ABD and ADC have the same area. Then ABD and ADC also have the same area. 1 Hence, BD and CD must have the same length, so BD = BC = 6. 2