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
Tom sent in the following neat solution that
uses some of the ideas suggested in the hints. He noticed that the
symmetry means you only need 'half' the diagram and that two radii
joining the centres of the circles to the tangents created useful
pairs of similar triangles.
Sue Liu of Madras College also sent a good
solution to this problem.
We start with Sue's proof that $XY$ is the
axis of symmetry of the whole shape.
Sue then goes on to prove in detail that
$ABDC$ is a rectangle. Her proof is an excellent piece of work
though a little longer than the proof below. The following proof
uses sines but it could equally well be written entirely in terms
of similar triangles.