HMMT 二月 2012 · 几何 · 第 8 题

英文解析

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 } .
   √ √
   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 ) xythird similarity gives 4 y = , so that x − 9 x + 12 = 0. It follows that x = , giving our 2
   answer.23