HMMT 十一月 2013 · 团队赛 · 第 6 题

英文解析

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 .
   ′ ′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