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Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

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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.

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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.

Calculating with Cosines

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

Try to make sense of each method one line at a time. The following mathematical topics may be useful:

Pythagoras' theorem

Cosines of angles greater than $90^{\circ}$

Distance between two points on a coordinate grid

Trigonometry in right-angled triangles