### 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

### Look Before You Leap

Can you spot a cunning way to work out the missing length?

### Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

# Two Triangles in a Square

##### Age 14 to 16 Challenge Level:

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