Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Find the answer

In $\triangle ABC, D$ and $E$ are the midpoints of $BC$ and $CA$ , respectively. $AD$ and $BE$ intersect at $G$ . Given that $GEC$ D is cyclic, $AB=41$ , and $AC=31$ , compute $BC$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

By Power of a Point, $\frac{2}{3}AD^{2}=AD \cdot AG=AE \cdot AC=\frac{1}{2} \cdot 31^{2}$ so $AD^{2}=\frac{3}{4} \cdot 31^{2}$ . The median length formula yields $AD^{2}=\frac{1}{4}\left(2AB^{2}+2AC^{2}-BC^{2}\right)$ whence $BC=\sqrt{2AB^{2}+2AC^{2}-4AD^{2}}=\sqrt{2 \cdot 41^{2}+2 \cdot 31^{2}-3 \cdot 31^{2}}=49$

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