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

We
received good solutions from several people. Let's start by looking
at the solution sent in by Tom of Wolgarston High School.
Firstly, let's consider the triangular problem. If instead of
coordinates we use vectors to describe the triangle, then we can
write ${\bf p}_1$, ${\bf p}_2$ and ${\bf p}_3$ to describe the
corners of the triangle (which we don't know yet) and ${\bf m}_1$,
${\bf m}_2$ and ${\bf m}_3$ to describe the midpoints of the
lines.

If we can find a way of expressing each vector ${\bf p}_1$, ${\bf
p}_2$ and ${\bf p}_3$ using just the vectors of the midpoints, then
we can locate the corners.

By the standard vector laws, the midpoint between any two points
descbribed by vectors is the average of those vectors. So we get
the following equations:

- ${\bf m}_1 = \frac{1}{2}({\bf p}_1 + {\bf p}_2)$
- ${\bf m}_2 = \frac{1}{2}({\bf p}_2 + {\bf p}_3)$
- ${\bf m}_3 = \frac{1}{2}({\bf p}_3 + {\bf p}_1)$

Now we solve the
simultaneous equations to find expressions for the corners. For
${\bf p}_1$ add equations 1 and 2 and then subtract equation 3.
This gives the following:

${\bf p}_1 = {\bf m}_1 + {\bf m}_2 -
{\bf m}_3 $

and similarly for ${\bf
p}_2$ and ${\bf p}_3$ we get:

${\bf p}_2 = {\bf m}_2 + {\bf m}_3 -
{\bf m}_1 $

${\bf p}_3 = {\bf m}_3 + {\bf m}_1 -
{\bf m}_2 $

Thus we have expressed all the triangle's vertices as expressions
of their midpoints.

See if
you can continue from here using the same method for
pentagons.
After
looking at pentagons Tom then went on to look at the quadrilaterals
and he found that he could no longer solve the simultaneous
equations in the same way. Ben Kenny noticed that if an arrangement
of midpoints produceded a quadrilateral then this quadrilateral was
not necessarily unique. For example:
Both quadrilaterals have the same midpoints but different
vertices.

Do you
notice anything special about when we can find a quadrilateral? Try
connecting the midpoints. Can you say anything interesting about
the inner quadrilateral? Try using Tom's method and see what
properties you might be able to gain by looking at the simultaneous
equations.