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Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

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Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

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Three Way Split

Take any point P inside an equilateral triangle. Draw PA, PB and PC from P perpendicular to the sides of the triangle where A, B and C are points on the sides. Prove that PA + PB + PC is a constant.

Tricircle

Age 14 to 16 Challenge Level:



Two circles (one large, one small) with a triangle overlaid.

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle.

A smaller circle touches the larger circle and two sides of the triangle.

If the small circle has radius 1 unit, find the radius of the larger circle.