HMMT 二月 2017 · 几何 · 第 2 题

英文解析

2. Let ABC be a triangle with AB = 13 , BC = 14, and CA = 15. Let ` be a line passing through twosides of triangle ABC . Line ` cuts triangle ABC into two figures, a triangle and a quadrilateral, thathave equal perimeter. What is the maximum possible area of the triangle?
   Proposed by: Sam Korsky
   1323
   Answer:
   There are three cases: ` intersects AB, AC , ` intersects AB, BC , and intersects AC, BC . These cases 26 are essentially identical, so let intersect segment AB at M and segment AC at N .
   Then the condition is equivalent to
   AM + M N + AN = M B + BC + CN + M N
   AM + AN = M B + CN + 15
   but AN + CN = 14 and AM + BM = 13, so that
   BM + CN = 27 − AM − AN = AM − AN − 15
   implying that AM + AN = 21.
   Now let ∠ BAC = θ for convenience, so that
   [ AM N ] = AM · AN · sin θ1
   21 122
   which is maximized when AM = AN = . Further we can easily calculate sin θ = (e.g. by LOC);
   2 13
   note that this is why the area is maximized in this case (we want to maximize sin θ , which is equivalentto maximizing θ , so θ should be opposite the largest side). Our answer is thus
   1 21 21 12 1323
   · · · =
   2 2 2 13 26
   Alternatively we could also calculate
   AM AN
   [ AM N ] = [ ABC ] · ·
   AB AC
   21 21
   2 2 = 84 · ·
   13 14
   which gives the same answer.