Triangle Incircle Iteration

Keep constructing triangles in the incircle of the previous triangle. What happens?

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

Stage: 4 Challenge Level:
Draw a circle with radius $1$ unit. Choose any four points on the circumference and join them together to form a quadrilateral.

Label the sides: $a$ is the shortest, $b$ is the next shortest, then $c$, and finally $d$ is the longest side (it is possible to have two sides of equal length).

What is the maximum length that the shortest side $a$ could be?

Side $b$ must be less than a certain value - what value?

What is the maximum length that the longest side $d$ could be? Is it possible for $c$ and $d$ both to be this maximum length?