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Bendy Quad

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

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

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

Circles in Quadrilaterals

Age 14 to 16
Challenge Level

Preveina from Crest Girls' Academy sent us some pictures to support her reasoning about some of the shapes in this problem:

A circle can be always fitted in a square touching all 4 sides since the sides of a square are all equal. This makes the circle touch each side of the square evenly.
A circle can never be fitted in to a rectangle touching all 4 sides because a rectangle has 2 long sides and 2 short sides. When you're trying to draw a circle that touches all 4 sides in a rectangle it'll turn out to be an oval, since there are 2 long sides.
A circle can never be fitted in to a parallelogram touching all 4 sides because a parallelogram has 2 long sides and 2 short sides just like a rectangle has.
A circle can also be fitted in to a kite touching all 4 sides.
The circle can be sometimes fitted into the trapezium touching all 4 sides depending on the length of the sides. If the non-parallel sides are too far apart, the circle becomes stretched into an oval.