### Circumspection

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.

### Lawnmower

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?

### 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:

Start with any triangle. Draw its inscribed circle (the circle which just touches each side of the triangle). Draw the triangle which has its vertices at the points of contact between your original triangle and its incircle. Now keep repeating this process starting with the new triangle to form a sequence of nested triangles and circles. What happens to the triangles?

If you wish, you can investigate this interactively:

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...)