HMMT 二月 2012 · 几何 · 第 8 题

HMMT February 2012 — Geometry — Problem 8

Hexagon ABCDEF has a circumscribed circle and an inscribed circle. If AB = 9, BC = 6, CD = 2, and EF = 4. Find { DE, F A } .

解析

Hexagon ABCDEF has a circumscribed circle and an inscribed circle. If AB = 9, BC = 6, CD = 2, and EF = 4. Find { DE, F A } . √ √ 9+ 33 9 − 33 Answer: { , } By Brianchon’s Theorem, AD, BE, CF concur at some point P . Also, it 2 2 follows from the fact that tangents from a point to a circle have equal lengths that AB + CD + EF = BC + DE + F A . Let DE = x , so that F A = 9 − x . Note that AP B ∼ EP D , BP C ∼ F P E , and CP D ∼ AP F . The second similarty gives BP/F P = 3 / 2, so that BP = 3 y, F P = 2 y for some y . From here, the first similarity gives DP = xy/ 3. Now, the √ 9 ± 33 (9 − x ) xy 2 third similarity gives 4 y = , so that x − 9 x + 12 = 0. It follows that x = , giving our 3 2 answer.