HMMT 十一月 2019 · 冲刺赛 · 第 16 题

HMMT November 2019 — Guts Round — Problem 16

[10] Equilateral 4 ABC has side length 6 . Let ω be the circle through A and B such that CA and CB are both tangent to ω. A point D on ω satisfies CD = 4 . Let E be the intersection of line CD with segment AB. What is the length of segment DE ?

解析

[10] Equilateral 4 ABC has side length 6 . Let ω be the circle through A and B such that CA and CB are both tangent to ω. A point D on ω satisfies CD = 4 . Let E be the intersection of line CD with segment AB. What is the length of segment DE ? Proposed by: Benjamin Qi 20 Answer: 13 Let F be the second intersection of line CD with ω . By power of a point, we have CF = 9, so DF = 5. [ ADB ] DE DE This means that = = . Now, note that triangle CAD is similar to triangle CF A , [ AF B ] EF 5 − DE F A CA 3 F B CB 3 so = = . Likewise, = = . Also, note that ∠ ADB = 180 − ∠ DAB − ∠ DBA = AD CD 2 BD CD 2 [ ADB ] AD · BD · sin 120 4 180 − ∠ CAB = 120, and ∠ AF B = 180 − ∠ ADB = 60. This means that = = . [ AF B ] F A · F B · sin 60 9 DE 4 20 Therefore, we have that = . Solving yields DE = . 5 − DE 9 13