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M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

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A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of grass exactly one foot wide and wishes to cut the entire lawn in parallel strips. What is the minimum number of strips the gardener must mow?

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Long Short

A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4. Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot be greater than sqrt2. Find a quadrilateral of this type for which s2 is approximately sqrt3and show that s2 is always less than sqrt3. Find a quadrilateral of this type for which s3 is approximately 2 and show that s2 is always less than 2. Find a quadrilateral of this type for which s4=2 and show that s4 cannot be greater than 2.

Triangle Incircle Iteration

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

This problem leads to a result which is easy to guess visually but not so easy to prove.
Geometrical, numerical, and algebraic ideas can all be used to reach a solution, and properties of averaging can also come out of the problem. Numerical patterns can be investigated using a spreadsheet.

Possible approach

Use the interactivity (or accurately construct some triangles with their inscribed circles) and see what happens to the angles in the nested triangles.In order to see why this is happening, it's important to make sure everyone knows that the centre of the inscribed circle is at the point where the angle bisectors of the original triangle meet, and that radii meet tangents at a right angle. This information can be used to write expressions for the three angles in the new triangle in terms of the original angles.

Students could now create a spreadsheet which allows them to input three angles which sum to 180 degrees and use their expressions to work out the three new angles. By continuing the sequence, the angles quickly converge.

Key questions

What seems to be happening to the angles in each new triangle that we draw?
How can we calculate the angles of each new triangle if we know the original angles?

Possible extension

Investigate the sequence $(90-x/2), 90-(90-x/2)/2$ and so on to explain why the angles converge to their limit.
The three new angles are each the mean of a pair of the original angles. In general, what happens if you keep finding the mean of pairs of numbers to give three new numbers?

Possible support

Work with numerical examples and try to explain the patterns formed.