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The Triangles Formed by the Endpoints and Midpoints of Cevians
Let ABC be a triangle with an interior point P. Let AP, BP, and CP intersect BC, AC, and AB 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 .



See problem 4.73 in Problems in Plane and Solid Geometry v.1 Plane Geometry by Viktor Prasolov.


"The Triangles Formed by the Endpoints and Midpoints of Cevians" from The Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/TheTrianglesFormedByTheEndpointsAndMidpointsOfCevians/
Contributed by: Jay Warendorff
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