HMMT November, 2023

Geometry Difficulty 4.8 AIME Prove it United States

Problem:

Suppose that point $D$ lies on side $BC$ of triangle $ABC$ such that $AD$ bisects $\angle BAC$ , and let $\ell$ denote the line through $A$ perpendicular to $AD$ . If the distances from $B$ and $C$ to $\ell$ are $5$ and $6$ , respectively, compute $AD$ .

Solution

Solution:

Let $\ell$ , the external angle bisector, intersect $BC$ at $X$ . By the external angle bisector theorem, $AB : AC = XB : XC = 5 : 6$ , so $BD : DC = 5 : 6$ by the angle bisector theorem. Then $AD$ is a weighted average of the distances from $B$ and $C$ to $\ell$ , namely
$\frac{6}{11} \cdot 5 + \frac{5}{11} \cdot 6 = \frac{60}{11}$

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