HMMT 二月 2008 · 冲刺赛 · 第 32 题

英文解析

32. [ 18 ] Cyclic pentagon ABCDE has side lengths AB = BC = 5 , CD = DE = 12 , and AE = 14 .
    Determine the radius of its circumcircle.
    √
    225 11
    ′ ′ ′
    Answer: Let C be the point on minor arc BCD such that BC = 12 and C D = 5 , and write
    ′ ′ ′ ′88
    AC = BD = C E = x, AD = y, and BD = z. Ptolemy applied to quadrilaterals ABC D, BC DE,
    and ABDE gives
    2 2
    x = 12 y + 5
    2 2
    x = 5 z + 12
    yz = 14 x + 5 · 12
    Then
    2 2 2 2 2 2
    ( x − 5 )( x − 12 ) = 5 · 12 yz = 5 · 12 · 14 x + 5 · 12 ,
    from which x − 169 x − 5 · 12 · 14 = 0 . Noting that x > 13 , the rational root theorem leads quickly to 3
    √ √
    5 · 12 · 15
    √
    the root x = 15 . Then triangle BCD has area 16 · 1 · 4 · 11 = 8 11 and circumradius R = =
    4 · 8 11
    √
    225 11
    .
    3 3 3 5 5 588