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 vertices B and D? What is the area of the square?

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?

Take a Square

Age 14 to 16 Challenge Level:

Cut off three congruent right angled isosceles triangles to produce a pentagon.

With two lines, cut the pentagon into three parts which can be rearranged into another square.