HMMT 二月 2016 · 几何 · 第 6 题

英文解析

6. Let ABC be a triangle with incenter I , incircle γ and circumcircle Γ. Let M , N , P be the midpointsof sides BC , CA , AB and let E , F be the tangency points of γ with CA and AB , respectively. Let U ,
   V be the intersections of line EF with line M N and line M P , respectively, and let X be the midpoint
   °
   ̂
   of arc BAC of Γ. Given that AB = 5, AC = 8, and ∠ A = 60 , compute the area of triangle XU V .
   Proposed by: Evan Chen
   √
   21 3
   Answer:
   Let segments AI and EF meet at K . Extending AK to meet the circumcircle again at Y , we see that 8
   X and Y are diametrically opposite, and it follows that AX and EF are parallel. Therefore the heightfrom X to U V is merely AK . Observe that AE = AF , so 4 AEF is equilateral; since M N, M P areparallel to AF, AE respectively, it follows that 4 M V U, 4 U EN, 4 F P V are equilateral as well. Then
   1 1 1 1 1
   M V = M P − P V = AC − F P = AC − AF + AP = AC − AF + AB = BC , since E, F are the
   2 2 2 2 2
   tangency points of the incircle. Since 4 M V U is equilateral, we have U V = M U = M V = BC .1
   Now we can compute BC = 7, whence U V = and 72
   √2
   AB + AC − BC 3 3
   °
   AK = · cos 30 = .
   2 2
   √
   21 3
   Hence, the answer is . 8