Geometry Level 5

Let $$ABC$$ be a triangle with $$BC = 20$$ and $$CA = 16$$, and let $$I$$ be its incenter. If the altitude from $$A$$ to $$BC$$, the perpendicular bisector of $$AC$$, and the line through I perpendicular to $$AB$$ intersect at a common point, then the length $$AB$$ can be written as $$m+\sqrt{n}$$ for positive integers $$m,n$$. What is $$100m+n?$$

Try my set JEE ADVANCED 2016

This problem is from the Spring OMO 2016

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