HMMT 十一月 2013 · 团队赛 · 第 6 题

HMMT November 2013 — Team Round — Problem 6

[ 6 ] Points A, B, C lie on a circle ω such that BC is a diameter. AB is extended past B to point B ′ ′ ′ and AC is extended past C to point C such that line B C is parallel to BC and tangent to ω at point ′ ′ D . If B D = 4 and C D = 6, compute BC .

解析

[ 6 ] Points A, B, C lie on a circle ω such that BC is a diameter. AB is extended past B to point B ′ ′ ′ and AC is extended past C to point C such that line B C is parallel to BC and tangent to ω at point ′ ′ D . If B D = 4 and C D = 6, compute BC . 24 ′ ′ Answer: Let x = AB and y = AC , and define t > 0 such that BB = tx and CC = ty . 5 √ ′ ′ 2 2 2 2 2 2 Then 10 = B C = (1 + t ) x + y , 4 = t (1 + t ) x , and 6 = t (1 + t ) y (by power of a point), so √ t (1+ t ) 13 52 t 13 2 2 2 2 2 2 52 = 4 + 6 = t (1 + t )( x + y ) gives = = = = ⇒ t = . Hence BC = x + y = 2 2 25 10 (1+ t ) 1+ t 12 10 10 24 = = . 1+ t 25 / 12 5