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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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Pythagoras for a Tetrahedron

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.

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Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.

Just Touching

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Two semi-circles are drawn on one side of a line segment. Another semi-circle touches them externally as shown in the diagram.

What is the radius of the circle that touches all three semi-circles?