### Two Trees

Two trees 20 metres and 30 metres long, lean across a passageway between two vertical walls. They cross at a point 8 metres above the ground. What is the distance between the foot of the trees?

### Golden Triangle

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

### Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

# The Eyeball Theorem

##### Stage: 4 and 5 Challenge Level:

In the interactive diagram below, the blue circle $C_1$ has a centre $X$ and fixed radius $R$, while the grey circle $C_2$ has a centre $Y$ and variable radius $r$. Two tangents to $C_2$ are drawn from $X$, cutting $C_1$ at points $A$ and $B$ and just touching $C_2$ at $P$ and $Q$. Tangents to $C_1$ are drawn from $Y$ cutting $C_2$ at points $C$ and $D$ and just touching $C_1$ at $R$ and $S$.

Click and drag the centres of the circles in the dynamic diagram below. As you change the distance between the centres of the circles and the radii what do you notice about the chords $AB$ and $CD$

Make and prove a conjecture about the chords $AB$ and $CD$.
 Created with GeoGebra

NOTES AND BACKGROUND

This dynamic image is drawn using Geogebra, free software and very easy to use. You can download your own copy of Geogebra from http://www.geogebra.org/cms/ together with a good help manual and Quickstart for beginners. You may be surprised at how easy it is to draw the dynamic diagram above for yourself.

Doing mathematics often involves observing and explaining properties of `invariance', that is, what remains the same when the rest of the pattern changes according to certain rules that can be described in mathematical terms. NRICH dynamic mathematics problems allow you to alter the diagrams and change some properties, so that you can observe what remains invariant. This may lead you to a conjecture that you can prove. Proving the result in the case of The Eyeball Theorem uses only similar triangles.