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Set Square

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Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

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Strange Rectangle

ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.

Flexi Quad Tan

Stage: 5 Challenge Level: Challenge Level:1
Consider any convex quadrilateral $Q$ made from four rigid rods with flexible joints at the vertices so that the shape of $Q$ can be changed while keeping the lengths of the sides constant. If the diagonals of the quadrilateral cross at an angle $\theta$ in the range $(0 \leq \theta < \pi/2)$, as we deform $Q$, the angle $\theta$ and the lengths of the diagonals will change.
Using the results of the two problems on quadrilaterals Diagonals for Area and Flexi Quads prove that the area of $Q$ is proportional to $\tan\theta$.