Pentagon Construction

Age 16 to 18 Challenge Level:
Here are a set of instructions for a construction. You could use pencil, ruler and compasses and draw it on plain paper, but you might like to use dynamic geometry software such as GeoGebra, which is free to download and use. If you have never used GeoGebra before, you might like to read this article first.

  1. Draw a circle with centre $A$ passing through a point $B$.
     
  2. Draw the line through $A$ and $B$. Mark the point where this intersects the circle again as $C$
     
  3. Draw in the line perpendicular to $AB$ passing through the point $A$. Mark the intersections with the circle as $D$ and $E$.
     
  4. Find the midpoint of $AB$ and mark it as $F$.
     
  5. Draw the circle with centre $F$ that passes through $D$. Mark the intersections with the line passing through $A$ and $B$ as $G$ and $H$.
     
  6. Draw the circle with centre $D$ that passes through $G$. Mark the intersections with the original circle as $I$ and $J$.
     
  7. Draw the circle with centre $D$ that passes through $H$. Mark the intersections with the original circle as $K$ and $L$.
     
  8. Join the vertices $D$, $I$, $K$, $L$ and $J$.


Can you prove that this construction forms a regular pentagon?


It may help to know the following identities:

\begin{eqnarray} \cos 72^\circ &=& \frac{\sqrt{5}-1}4\\
\cos 144^\circ &=& \frac{-1-\sqrt{5}}4
\end{eqnarray}