HMMT 二月 2002 · 冲刺赛 · 第 53 题

英文解析

53. ABC is a triangle with points E, F on sides AC, AB , respectively. Suppose that
    BE, CF intersect at X . It is given that AF/F B = ( AE/EC ) and that X is the midpoint 2
    of BE . Find the ratio CX/XF .
    Solution: Let x = AE/EC . By Menelaus’s theorem applied to triangle ABE and line
    CXF ,
    AF BX EC x 2
    1 = · · = .
    F B XE CA x + 1
    √
    Thus, x = x + 1, and x must be positive, so x = (1 + 5) / 2. Now apply Menelaus to 2
    triangle ACF and line BXE , obtaining
    AE CX F B CX x
    1 = · · = · ,
    EC XF BA XF x + 12
    √
    2 2
    so CX/XF = ( x + 1) /x = (2 x − x ) /x = 2 x − 1 = 5 .