# Points on Leg of Right Triangle with Distance One

Geometry Level 4

Suppose $$\triangle ABC$$ is a right triangle with $$\angle A=90^\circ$$, $$BC=18$$, and $$AC > AB$$. Let $$M$$ be the midpoint of line segment $$\overline{BC}$$. Suppose there exist points $$D_1$$ and $$D_2$$ on $$AC$$ such that $$\overline{BD_1}\perp\overline{D_1M}$$, $$\overline{BD_2}\perp\overline{D_2M}$$, and $$D_1D_2=1$$. The ratio $$\tfrac{AC}{AB}$$ can be written in the form $$\sqrt{\tfrac mn}$$, where $$m$$ and $$n$$ are coprime positive integers. What is $$m+n?$$

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