HMMT 二月 2016 · 几何 · 第 4 题

英文解析

4. Let ABC be a triangle with AB = 3, AC = 8, BC = 7 and let M and N be the midpoints of AB and
   AC , respectively. Point T is selected on side BC so that AT = T C . The circumcircles of triangles
   BAT , M AN intersect at D . Compute DC .
   Proposed by: Evan Chen
   √
   7 3
   Answer:
   We note that D is the circumcenter O of ABC , since 2 ∠ C = ∠ AT B = ∠ AOB . So we are merely looking 3
   √
   for the circumradius of triangle ABC . By Heron’s Formula, the area of the triangle is 9 · 6 · 1 · 2 =
   √
   √
   abc 3 · 8 · 7 7 3
   √
   6 3, so using the formula = K , we get an answer of = . Alternatively, one can compute
   4 R 3
   4 · 6 3
   °
   the circumradius using trigonometric methods or the fact that ∠ A = 60 .