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Right triangle with a circle

Source: 2009 AIME I #12

March 18, 2009
ratiogeometryperimeterinradiustrapezoidAMCAIME

Problem Statement

In right △ABC \triangle ABC with hypotenuse AB‾ \overline{AB}, AC \equal{} 12, BC \equal{} 35, and CD‾ \overline{CD} is the altitude to AB‾ \overline{AB}. Let ω \omega be the circle having CD‾ \overline{CD} as a diameter. Let I I be a point outside △ABC \triangle ABC such that AI‾ \overline{AI} and BI‾ \overline{BI} are both tangent to circle ω \omega. The ratio of the perimeter of △ABI \triangle ABI to the length AB AB can be expressed in the form mn \displaystyle\frac{m}{n}, where m m and n n are relatively prime positive integers. Find m\plus{}n.
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