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In the diagram 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 that
(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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