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Medians as diameters and a known Feuerbach-type result

Source: Romanian TST 5 2008, Problem 2

June 13, 2008
geometrygeometry proposed

Problem Statement

Let ABC ABC be a triangle and let Ma \mathcal{M}_{a}, Mb \mathcal{M}_{b}, Mc \mathcal{M}_{c} be the circles having as diameters the medians ma m_{a}, mb m_{b}, mc m_{c} of triangle ABC ABC, respectively. If two of these three circles are tangent to the incircle of ABC ABC, prove that the third is tangent as well.
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