### Set Square

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

### Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

### Strange Rectangle

ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.

# Strange Rectangle 2

##### Stage: 5 Challenge Level:

David provided us with his solution:

I firstly used the angles calculated in *** and put them on the diagram. Next I used the fact that $\sin30=1/2$ (this isn't too difficult to prove), to find that $z=30$ degrees. From this, I could work out all the other angles and fill them in:

from this all the sine, cosine and tangent values can be read off.

For the final part, I noticed when

$x = \sqrt{2} + 1$ and $y = 1$ then $SD$ and $AS$ are the same length. This makes $SDR$ and $SAP$ similar triangles, and so the angle $DSR$ is $22.5$ degrees. So we get:

From which all the sine, cosine and tangent values of $22.5$ degrees can be obtained.