### Just Opposite

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?

### 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

### Compare Areas

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

# Two Triangles in a Square

##### Stage: 4 Challenge Level:

Given that $ABCD$ is a square, $M$ is the mid point of $AD$ and $PC$ is perpendicular to $MB$, prove $DP = DC$.