A parallelogram is inscribed in a quadrilateral. What fraction is red?
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Solution
The red fraction is 1/2.
First consider the quadrilateral connecting all four midpoints of the original quadrilateral. This is a parallelogram and it is known as the Varignon parallelogram. Its area is half the containing quadrilateral’s ABCD.
Now assume a non-midpoint vertex G on side AD and mirror it in the midpoint of EF to obtain the other vertex G’ of a parallelogram. This vertex G’ is on the other side BC if and only if BC is parallel to AD (and EF). In that case GEG’F also has areal fraction 1/2 of ABCD.
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