### Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

### Circumspection

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

# Semi-detached

### Why do this problem?

This problem could work well as a 'poster' - a visual challenge placed where students will see it. Or presented at the end of a lesson as something to try to solve.

### Possible approach

This problem can worked well as something short and closed, but there is also an opportunity to invite questions which open up beyond the initial challenge. Finding the area of a square in a quadrant or squares fitted in the space between either of the two squares in the main problem and the circle.

### Key questions

• What is the challenge and how might you start ?
• How did you do it ? Can you explain ?
• How might you extend this problem ?
• Can you calculate the area of squares fitted into other places within this diagram ?

### Possible support

Tilted Squares could be an excellent and accessible challenge for slightly less experienced students.

### Possible extension

The approach suggested above indicates one route to extension within this context, or for another challenge fitting squares into shapes try Squirty. The problem Semi-square offers another opportunity to work out areas of squares inside circles.