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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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Compare Areas

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

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Zig Zag

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

Just Opposite

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

$A$ and $C$ are the opposite vertices of a square $ABCD$, and have coordinates $(a,b)$ and $(c,d)$, respectively.

What are the coordinates of the other two vertices?

What is the area of the square?

How generalisable are these results?