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## 'Pentagon' printed from http://nrich.maths.org/

It is often advisable to start with a simple case of the same
problem. Here you might like first to think of the same problem for
a triangle where you are given the midpoints of the sides and have
to find the vertices. The problem suggests doing this with
mid-point coordinates $(6, 0)$, $(6.5, 2)$, $(7.5, 1)$. Write down
and solve some simultaneous equations and find the vertices.

Then if you choose any set of 3 mid-points could you find the
vertices?

You can then extend the method to pentagons.

Then consider the differences between cases of polygons with an odd
number of sides and polygons with an even number of sides.

Compare this to the problem
Polycircles.

Vassil Vassilev from Lawnswood High School, has an idea for
constructing pentagons from the midpoints of edges based on nested
pentagons, and 5 pointed stars within them, which are all
enlargements of each other. You might like to play with this
idea.