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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