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