If you are given the coordinates of the midpoints of the edges of a pentagon can you find the coordinates of the vertices of the pentagon?

An extension of the problem could be: given any number of points in a plane can you construct a polygon with the given points as midpoints of the edges?

Let's explore this!

In the diagram, the moveable red points labelled $A$ to $E$, are the midpoints of the line segments passing through them.

For any given points $A$, $B$, $C$, $D$ and $E$, can you always drag the point $F$ to coincide with the point $K$ making a pentagon with the given points at the midpoints of the edges?

If so, find the positions of the vertices $F$ (aka $K$), $G$, $H$, $I$ and $J$.

If you can't see how to find the five vertices try the next part of the question.

A good problem solving strategy is to start with a simple case. What about three points?

Again the moveable red points labelled $A$, $B$ and $C$ are the midpoints of the line segments passing through them. Can you always make a triangle with the given points at the midpoints of the edges?

Try a numerical example. Calculate the coordinates of the vertices of the triangle with $(6, 0)$, $(6.5, 2)$ and $(7.5, 1)$ as the midpoints of the edges?

Find the vertices of the triangle with $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$ as the midpoints of the edges?

Now can you use the same method for a pentagon?

What about quadrilaterals? Given four points can we always find a quadrilateral with the given points as midpoints of the edges?

In the diagram, the moveable red points are the midpoints of the line segments passing through them.

Is it ever impossible to find a quadrilateral with the given points as midpoints of the edges?

When you can find a solution, that is you can find a quadrilateral when given certain midpoints of the edges, is it a unique solution?

If you can find more than one solution for a particular set of midpoints, how many solutions do you think there are?

Compare this problem to the problem Polycircles.

Created with GeoGebra

NOTES AND BACKGROUND

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