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Triangle Incircle Iteration

Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this process starting with T2 to form a sequence of nested triangles and circles. What happens to the triangles? You may like to investigate this interactively on the computer or by drawing with ruler and compasses. If the angles in the first triangle are a, b and c prove that the angles in the second triangle are given (in degrees) by f(x) = (90 - x/2) where x takes the values a, b and c. Choose some triangles, investigate this iteration numerically and try to give reasons for what happens. Investigate what happens if you reverse this process (triangle to circumcircle to triangle...)

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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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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.


Stage: 4 Challenge Level: Challenge Level:1

Tony Cardell, has sent in this solution, which gives the correct answer. We're still not quite convinced: how does he know that the strips are parallel to the longer side? If anyone can explain this, we'll add their explanation here.

We must find the maximum distance between a pair of opposite sides (it doesn't matter which since this is a kite). If we extend the two sides (coloured blue), the intersection is on the same side of the kite as the equilateral triangle, as shown. So the maximum distance between the two sides is the red line.


Now look at the isosceles triangle part of the kite. One formula for the area of a triangle with sides a,b,c says that if s is the semi-perimeter, (a+b+c)/2, then the area is the square root of s(s-a)(s-b)(s-c). The isosceles triangle has sides 169, 169, 130, so s=234 and the area is 10140 square feet. But we also know that the area is 1/2 x base x height, so the height we want is 10140 x 2 / 169=120. So 120 foot-wide strips will be needed.