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?

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.

Sangaku

The square ABCD is split into three triangles by the lines BP and CP. Find the radii of the three inscribed circles to these triangles as P moves on AD.

So Big

Stage: 5 Challenge Level:

 A circle of radius $r$ is drawn inside a triangle so that it just touches each of the three sides as shown in the diagram. The three corners and points where the circle touches have been labelled $A$ to $F$. One side of the triangle is divided into segments of length $a$ and $b$ by the inscribed circle. However, we are not told which of the three sides is divided in this way. From this information we can find an expression for the area of the triangle. Prove that the area of the triangle is: $$\frac{abr(a+b)}{ab-r^2}.$$