HMMT 十一月 2013 · 团队赛 · 第 4 题

HMMT November 2013 — Team Round — Problem 4

[ 4 ] Consider triangle ABC with side lengths AB = 4, BC = 7, and AC = 8. Let M be the midpoint of segment AB , and let N be the point on the interior of segment AC that also lies on the circumcircle of triangle M BC . Compute BN . ◦

解析

[ 4 ] Consider triangle ABC with side lengths AB = 4, BC = 7, and AC = 8. Let M be the midpoint of segment AB , and let N be the point on the interior of segment AC that also lies on the circumcircle of triangle M BC . Compute BN . √ √ 2 2 2 210 105 4 +8 − 7 4 √ Answer: OR Let ∠ BAC = θ . Then, cos θ = . Since AM = = 2, and power 4 2 · 4 · 8 2 2 2 2 · 4 of a point gives AM · AB = AN · AC , we have AN = = 1, so N C = 8 − 1 = 7. Law of cosines on 8 triangle BAN gives 2 2 2 4 + 8 − 7 16 + 15 15 105 2 2 2 BN = 4 + 1 − 2 · 4 · 1 · = 17 − = 15 − = , 2 · 4 · 8 8 8 8 √ 210 so BN = . 4 ◦