Concurrency via Midpoints
Let ABC be a triangle and P be an internal point. Let AP, BP, and CP intersect the sides BC, CA, and AB in A', B', and C'. Let L, M, and N be the midpoints of the sides of ABC and L', M', and N' be the midpoints of the sides of A'B'C'. Then LL', MM', and NN' are concurrent.
See problem 20 in Classical Theorems in Plane Geometry.