Intersection of an Altitude and a Line through the Incenter

Let ABC be a triangle with incenter I and inradius . Let M be the midpoint of BC, H the foot of the altitude drawn from A, and E the intersection of MI with AH. Then .

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See problem 5.7 in V. Prasolov, Problems in Plane and Solid Geometry, Vol. 1, Plane Geometry [PDF], (D. Leites, ed. and trans.).
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