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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:
 .
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FACTS AND HINTS:
Geometry problem solving is one of the most
challenging skills for students to learn. When a
problem requires auxiliary construction, the
difficulty of the problem increases drastically,
perhaps because deciding which construction to
make is an ill-structured problem. By
“construction,” we mean adding geometric figures
(points, lines, planes) to a problem figure that
wasn’t mentioned as "given."
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:
Proposition:
The area of a triangle equals
one-half the product of the length of a side and the length of
the altitude to that side.
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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