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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
Triangle Midpoints
You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
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.
Age 14 to 16 Challenge Level:
If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units....
What is the area of the triangle?
What other heights and perimeters less than or equal to 100 will give you a whole number area?
NRICH is part of the family of activities in the Millennium Mathematics Project.