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