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In the figure below, given a
quadrilateral ABCD of area S. E, F, G, and H are the centroids of the
triangles ABC, BCD, ACD, and ABD respectively. S1 is the area of
the quadrilateral EFGH. Prove the
following:
1. EF, FG, GH, and EH are parallel to
AD, AB, BC, and CD respectively.
2. S = 9S1
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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. CENTROID:
The geometric centroid (center of mass) or barycenter of the polygon vertices of a triangle is the point G which is also the intersection of the triangle's three triangle medians.
Proposition: The centroid divides each of the medians in the ratio 2:1
2. PROPORTIONAL SEGMENTS:
Proposition 1: If a line is parallel to one side of a
triangle, then it divides the other two sides proportionally.
Proposition 2: If a line divides two sides of a triangle
proportionally, it is parallel to the third side. (Converse of
-proposition 1.)
3. SIMILAR POLYGONS - Ratio of Areas:
Proposition: If two polygons are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
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