HMMT 二月 2002 · 几何 · 第 10 题

HMMT February 2002 — Geometry — Problem 10

Let 4 ABC be equilateral, and let D, E, F be points on sides BC, CA, AB respectively, with F A = 9 , AE = EC = 6 , CD = 4. Determine the measure (in degrees) of ∠ DEF . 1

解析

Let 4 ABC be equilateral, and let D, E, F be points on sides BC, CA, AB respec- tively, with F A = 9 , AE = EC = 6 , CD = 4. Determine the measure (in degrees) of ∠ DEF . Solution: 60 . Let H, I be the respective midpoints of sides BC, AB , and also extend CB and EF to intersect at J . By equal angles, 4 EIF ∼ 4 JBF . However, BF = 12 − 9 = ∼ 3 = 9 − 6 = IF , so in fact 4 EIF 4 JBF , and then JB = 6. Now let HI intersect EF = at K , and notice that 4 EIK ∼ 4 JHK ⇒ IK/HK = EI/JH = 6 / 12 = 1 / 2 ⇒ HK = 4, ◦ since IK + HK = HI = 6. Now consider the 60 rotation about E carrying triangle CHE ◦ to triangle HIE ; we see that it also takes D to K , and thus ∠ DEF = ∠ DEK = 60 . 3