HMMT 二月 2009 · 几何 · 第 7 题

英文解析

7. [ 5 ] In 4 ABC , D is the midpoint of BC , E is the foot of the perpendicular from A to BC , and F isthe foot of the perpendicular from D to AC . Given that BE = 5, EC = 9, and the area of triangle
   ABC is 84, compute | EF | .
   √
   √
   6 37 21
   Answer: , 7585
   5 205
   Solution: There are two possibilities for the triangle ABC based on whether E is between B and
   C or not. We first consider the former case.
   We find from the area and the Pythagorean theorem that AE = 12, AB = 13, and AC = 15. We can
   √
   then use Stewart’s theorem to obtain AD = 2 37.
   Since the area of 4 ADC is half that of ABC , we have AC · DF = 42, so DF = 14 / 5. Also,1
   DC = 14 / 2 = 7 so ED = 9 − 7 = 2.2
   √
   Notice that AEDF is a cyclic quadrilateral. By Ptolemy’s theorem, we have EF · 2 37 = (28 / 5) · 12 +
   √
   6 37
   2 · (54 / 5). Thus EF = as desired.
   The latter case is similar.5