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?
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.
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
This dynamic image is drawn using Geogebra, free software and very easy to use. You can download your
own copy of Geogebra from
together with a good help manual and
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