The GeoGebra applet below shows steps that can be used to construct a regular pentagon. Press the play/pause button, or use the arrows to go through one step at a time.

Copy this straight edge and compass construction. Can you explain why it produces a regular pentagon?

The description of the construction below, and the information in the notes, should help you to explain the construction.

Here are the steps:

1. Draw a circle $C_1$ centre $O$ diameter $PQ$.

*The circle $C_1$ has radius 1 unit; what is its equation?*

2. Draw the perpendicular bisector of $PQ$ cutting $PQ$ at $O$ and $C_1$ at $A$ and $Y$.

3. Draw perpendicular bisectors of $PO$ and $OQ$ cutting $PQ$ at $R$ and $S$.

*Find the length $YS$*

4. Draw circles $C_2$ and $C_3$ centres $R$ and $S$ and radii $RO$ and $SO$.

5. Join $R$ and $S$ to the point $Y$ cutting $C_2$ at $T$ and $U$ and $C_3$ at $V$ and $W$.

6. Draw circle $C_4$ centre $Y$ radius $YW=YU$ cutting $C_1$ at $D$ and $C$.

*What is the equation of $C_4$? Find the value of $y$ at the intersection of $C_1$ and $C_4$* .

7. Draw circle $C_5$ centre $Y$ radius $YT=YV$ cutting $C_1$ at $E$ and $B$.

*What is the equation of $C_5$* ?

Find the value of $y$ at the intersection of $C_1$ and $C_5$.

At $B$ and $E$ $x^2 + y^2 +2y +1 = 2y + 2 = (3 + \sqrt 5)/2$ so

8. Join $AB$, $BC$, $CD$, $DE$, $EA$.

How could you adapt this construction to produce a regular decagon?