HMMT February, 2016

Geometry Difficulty 5.2 AIME, harder Prove it United States

Problem:

Let $ABC$ be a triangle with $AB = 3$ , $AC = 8$ , $BC = 7$ and let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC}$ , respectively. Point $T$ is selected on side $BC$ so that $AT = TC$ . The circumcircles of triangles $BAT$ , $MAN$ intersect at $D$ . Compute $DC$ .

Solution

Solution:

We note that $D$ is the circumcenter $O$ of $ABC$ , since $2 \angle C = \angle ATB = \angle AOB$ . So we are merely looking for the circumradius of triangle $ABC$ . By Heron's Formula, the area of the triangle is $\sqrt{9 \cdot 6 \cdot 1 \cdot 2} = 6\sqrt{3}$ , so using the formula $\frac{abc}{4R} = K$ , we get an answer of $\frac{3 \cdot 8 \cdot 7}{4 \cdot 6\sqrt{3}} = \frac{7\sqrt{3}}{3}$ . Alternatively, one can compute the circumradius using trigonometric methods or the fact that $\angle A = 60^\circ$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.