HMMT 十一月 2022 · 冲刺赛 · 第 32 题
HMMT November 2022 — Guts Round — Problem 32
题目详情
- [17] Suppose point P is inside triangle ABC . Let AP, BP, and CP intersect sides BC, CA, and AB at 1 1 1 points D, E, and F , respectively. Suppose ∠ AP B = ∠ BP C = ∠ CP A , P D = , P E = , and P F = . 4 5 7 Compute AP + BP + CP .
解析
- [17] Suppose point P is inside triangle ABC . Let AP, BP, and CP intersect sides BC, CA, and AB 1 1 at points D, E, and F , respectively. Suppose ∠ AP B = ∠ BP C = ∠ CP A , P D = , P E = , and 4 5 1 P F = . Compute AP + BP + CP . 7 Proposed by: Rishabh Das 19 Answer: 12 Solution: The key is the following lemma: ◦ Lemma: If ∠ X = 120 in △ XY Z , and the bisector of X intersects Y Z at T , then 1 1 1
- = . XY XZ XT Proof of the Lemma. Construct point W on XY such that △ XW T is equilateral. We also have T W ∥ XZ . Thus, by similar triangles, XT Y T XT = = 1 − , XZ Y X XY implying the conclusion. Now we can write 1 1
- = 4 , P B P C 1 1
- = 5 , and P C P A 1 1
- = 7 . P A P B 1 1 1 19 From here we can solve to obtain = 4 , = 3 , = 1 , making the answer . P A P B P C 12