Given a way of finding the midpoint of any segment (and connecting points to make lines, and intersecting lines to get points), construct the average of 5 points, using only 4 midpoint constructions. (Shown is a method using 5.)
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Solution
By using the centroid of the CDE triangle as a connection point, we reduce the number of midpoints used to 4. In this case either EJ or DL can be used as the intersection. I note that in any case, intersection lines at P must be a linear combination of all five points.
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