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Set Square

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

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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 2

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

Strange Rectangle

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

To crack this Tough Nut look for pairs of similar triangles and angles adding up to $90$ degrees and this leads to $ PQRS $ being a cyclic quadrilateral with $ SQ $ as diameter. You need to know that opposite angles of a cyclic quadrilateral add up to $180$ degrees and the angle at the centre of a circle is twice the angle at the circumference subtended by the same arc. Call the centre of the circle $ O $ the use the converse of Pythagoras' Theorem to find the angle $ POR $ . The rest follows.