HMMT 十一月 2022 · 冲刺赛 · 第 20 题

HMMT November 2022 — Guts Round — Problem 20

[11] Let △ ABC be an isosceles right triangle with AB = AC = 10. Let M be the midpoint of BC and N the midpoint of BM . Let AN hit the circumcircle of △ ABC again at T . Compute the area of △ T BC.

解析

[11] Let △ ABC be an isosceles right triangle with AB = AC = 10. Let M be the midpoint of BC and N the midpoint of BM . Let AN hit the circumcircle of △ ABC again at T . Compute the area of △ T BC. Proposed by: Andrew Gu Answer: 30 Solution: Note that since quadrilateral BACT is cyclic, we have ◦ ∠ BT A = ∠ BCA = 45 = ∠ CBA = ∠ CT A. ◦ Hence, T A bisects ∠ BT C , and ∠ BT C = 90 . By the angle bisector theorem, we then have BT BN 1 = = . T C N C 3 By the Pythagorean theorem on right triangles △ T BC and △ ABC , we have 2 2 2 2 2 10 BT = BT + T C = AB + AC = 200 , 2 so BT = 20. Note that the area of △ T BC is 2 BT · T C 3 · BT = , 2 2 so our answer is then 3 3 2 · BT = · 20 = 30 . 2 2