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

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

Age 14 to 16 Challenge Level:

$A$ 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?