HMMT 二月 2006 · 冲刺赛 · 第 19 题

英文解析

19. Let ABC be a triangle with AB = 2 , CA = 3 , BC = 4. Let D be the point diametrically opposite A on the circumcircle of ABC , and let E lie on line AD such that D isthe midpoint of AE . Line l passes through E perpendicular to AE , and F and G are the intersections of the extensions of AB and AC with l . Compute F G .
    1024
    Answer:
    √45
    3 15
    Solution: Using Heron’s formula we arrive at [ ABC ] = . Now invoking the rela-
    ( )4
    abc 2 2 · 3
    tion [ ABC ] = where R is the circumradius of ABC , we compute R = =
    4 R [ ABC ]2
    . Now observe that ∠ ABD is right, so that BDEF is a cyclic quadrilateral. Hence 64
    512 51215
    AB · AF = AD · AE = 2 R · 4 R = . Similarly, AC · AG = . It follows that
    15 15
    BCGF is a cyclic quadrilateral, so that triangles ABC and AGF are similar. Then
    AF 512 1024
    F G = BC · = 4 · =
    AC 2 · 15 · 3 45