# Online Geometry Problem 89. Area of Triangle and Quadrilateral, Midpoints of Diagonals, Median of a triangle. Level: High School, College, Mathematics Education

 In the figure below, given a triangle AED, M and N are the midpoints of cevians AC and DB respectively. If S1, S2, and S3 are the areas of the triangles EBM, ECN, and BEC respectively, prove that: .       "A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such expert experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime." George Polya, 1944    HINTS: 1. CEVIAN: A Cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension).   2. AREA OF A TRIANGLE: Median Area Fact: A median divides the triangle into two triangles of equal area.   3. Mid-Segment or Midline of a Triangle Theorem: If a line MN joins the midpoints of two sides of a triangle, then it is parallel to the third side and its length is one-half the length of the third side.
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