HMMT 二月 2006 · 冲刺赛 · 第 19 题

HMMT February 2006 — Guts Round — Problem 19

[8] Let ABC be a triangle with AB = 2 , CA = 3 , BC = 4. Let D be the point diametrically opposite A on the circumcircle of ABC , and let E lie on line AD such that D is the midpoint of AE . Line l passes through E perpendicular to AE , and F and G are the intersections of the extensions of AB and AC with l . Compute F G .

解析

Let ABC be a triangle with AB = 2 , CA = 3 , BC = 4. Let D be the point diametri- cally opposite A on the circumcircle of ABC , and let E lie on line AD such that D is the midpoint of AE . Line l passes through E perpendicular to AE , and F and G are the intersections of the extensions of AB and AC with l . Compute F G . 1024 Answer: 45 √ 3 15 Solution: Using Heron’s formula we arrive at [ ABC ] = . Now invoking the rela- 4 ( ) abc 2 2 · 3 tion [ ABC ] = where R is the circumradius of ABC , we compute R = = 2 4 R [ ABC ] 64 . Now observe that ∠ ABD is right, so that BDEF is a cyclic quadrilateral. Hence 15 512 512 AB · AF = AD · AE = 2 R · 4 R = . Similarly, AC · AG = . It follows that 15 15 BCGF is a cyclic quadrilateral, so that triangles ABC and AGF are similar. Then AF 512 1024 F G = BC · = 4 · = AC 2 · 15 · 3 45