HMMT 二月 2016 · 几何 · 第 4 题

HMMT February 2016 — Geometry — Problem 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 .

解析

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: 3 We note that D is the circumcenter O of ABC , since 2 ∠ C = ∠ AT B = ∠ AOB . So we are merely looking √ 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 .