The Triangles Formed by the Endpoints and Midpoints of Cevians

Let ABC be a triangle and P be a point. Let AP, BP, and CP intersect BC, AC, and AB or their extensions at A', B', and C', respectively. Let A'', B'', and C'' be the midpoints of AA', BB', and CC'. Let and be the areas of A'B'C' and A''B''C''. Then .

[1] V. Prasolov. Problems in Plane and Solid Geometry, Vol. 1: Plane Geometry (D. Leites, ed. and trans.), Problem 1.35. (Jul 16, 2010) www.students.imsa.edu/~tliu/Math/planegeo.pdf.