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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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