### Kissing

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

### Golden Construction

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

### A Rational Search

Investigate constructible images which contain rational areas.

# Pentabuild

##### Age 16 to 18 Challenge Level:

 The film strip shows the steps in the construction of the regular pentagon. Copy this straight edge and compass construction for yourself and 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.
Full Screen Version
If you can see this message Flash may not be working in your browser
Please see http://nrich.maths.org/techhelp/#flash to enable it.
Here are the steps shown in the film sequence:

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 would you adapt this construction to produce a regular decagon?