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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?

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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.

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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: Challenge Level:2 Challenge Level:2

Why do this problem?

At first sight there might not seem to be enough information given in this question to find a solution. The first step is to interpret what is known from the geometry given the inscribed circle, and then to evaluate that information in terms of how it might be used to express what is known in terms of the three variables $a$, $b$ and $r$ and how that might be used to give an expression for the area of the triangle.

One of the methods of solving this problem is an application of the formula for tan$(A + B)$ combined with the formula for the area of a triangle.

Possible approach

Set this as an exercise in applying the formula for $tan(A + B)$ or better still, if you want the students to revise their trigonometry, you can make it a more challenging problem solving activity and leave it to the students to decide what they have to use.

Key questions

Can we make a list of everything that is known from the information given?

Can we see a way of using the information given to find the area of the triangle?

Can we add lines to make pairs of congruent triangles?

Can we find the area of the whole triangle in terms of these smaller congruent triangles?

What can we say about the angles at the centre of the circle?

If three of the angles at the centre add up to $180$ degrees what can we say about the tangents of these angles?