Reflecting a Point through the Midpoints of a Triangle's Sides

Let ABC be a triangle and let A', B', and C' be symmetric to P with respect to the midpoints of BC, CA, and AB, respectively. Then AA', BB', and CC' are concurrent.

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See problem 5.43 in V. Prasolov, Problems in Plane and Solid Geometry, Vol. 1, Plane Geometry [PDF], (D. Leites, ed. and trans.).
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