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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
What are the limitations on the lengths of the sides of any triangle?
Look at different cases.
If the base is a 4 by 5 by 6 triangle, what size can the other faces be?
How many different tetrahedra can have as its base a 4 by 5 by 6 triangle?
Can you find 8 or more possible tetrahedra?
How will you know when you have them all?
NRICH is part of the family of activities in the Millennium Mathematics Project.