HMMT 十一月 2010 · 冲刺赛 · 第 31 题

英文解析

31. [ 20 ] What is the perimeter of the triangle formed by the points of tangency of the incircle of a 5-7-8
    triangle with its sides?
    Guts Round
    √
    9 21
    Answer: + 3 Let 4 ABC be a triangle with sides a = 7, b = 5, and c = 8. Let the incircle of
    4 ABC be tangent to sides BC , CA , and AB at points D , E , and F . By the law of cosines (using the 7
    2 2 2
    b + c − aform cos( A ) = ), we have
    2 bc
    2 2 2
    8 + 5 − 7 1
    cos( A ) = =
    2(5)(8) 2
    2 2 2
    8 + 7 − 5 11
    cos( B ) = =
    2(7)(8) 14
    2 2 2
    5 + 7 − 8 1
    cos( C ) = =
    2(7)(5) 7
    Now we observe that AEF , BDF , and CDE are all isosceles. Let us call the lengths of the legs ofthese triangles s , t , and u , respectively. Then we know that s + t = 8, t + u = 7, and u + s = 5, sos = 3, t = 5, and u = 2.
    Our final observation is that an isosceles angle with legs of length l and whose non-equal angle is θ
    √
    has a base of length l 2(1 − cos( θ )). This can be proven using the law of cosines or the Pythagoreantheorem.
    Using this, we can calculate that
    √
    DE = 2 2(1 − cos( C ))
    √
    = 212
    √7
    EF = 3 2(1 − cos( A )) = 3
    √
    F D = 5 2(1 − cos( B ))
    √
    = 5 ,3
    and then 7
    √ √
    12 3
    DE + EF + F D = 2 + 3 + 5
    7 7
    √
    = 3 + 93
    √7 = 3 + 9 .21
    a b 7