Mean Geometrically

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

So Big

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

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.

Vecten

Stage: 5 Challenge Level:

This diagram gives you many big hints.

Can you find reasons and explanations for how it all fits together as it appears to do?

It is important to bear in mind that the angles around a point add up to $360^o$ and the angles on a straight line make $180^o$.

When you have found the angles in the quadrilaterals does this suggest that the quadrilaterals are a special type?

See how $3$ copies of triangle $D$ and two chosen from triangles $r, s$ and $t$ fit into the quadrilaterals like pieces of a jigsaw. Can you explain why it works?