### Fixing It

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

### OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

### 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?

### Why do this problem?

This problem introduces students to a surprising result which holds true for all quadrilaterals, although in the problem we invite them to explore convex quadrilaterals first. The GeoGebra interactivity provides a hook to engage students, and we hope the result will be intriguing to students, encouraging their curiosity and giving them a desire to persevere until they have proved the result.

### Possible approach

This problem featured in an NRICH Secondary webinar in March 2022.

These printable cards for sorting may be useful: Shape of PQRS, Area of PQRS

You may wish to begin by inviting students to draw their own quadrilaterals on dotty paper, which you can print off from our Printable Resources page. Once they have generated a few examples, invite them to explain anything they have noticed. This is a really good opportunity for discussion about the examples they chose - did their original quadrilaterals have any particular symmetry properties that might have caused the results they are finding? Can they find an example that doesn't work?

At this point, you might like to encourage students to explore using dynamic geometry, either by using the interactivity in the problem, the interactivity created by Alison Kiddle, or constructing their own using software such as GeoGebra, which is free to use.

Diagrams such as this one might be useful for helping students to construct a proof:

Give students some time to have a go at the problem. While they are working, circulate and see the methods they are trying. As well as considering the shape formed, students could also calculate the area of the original quadrilateral and the new one formed by the midpoints, and look for a relationship.

After a while, bring the class together again and acknowledge that the proofs may not be immediately obvious.

"I've been given two different proofs (one for the shape and one for the area properties), but unfortunately both have been jumbled up. Can you put the statements in the right order to build a logical argument?"

Hand out envelopes with the statements for the Shape proof (or direct students to the interactive proof sorter) and the Area proof (or direct students to the interactive proof sorter)
"With your partner, make sense of each step and put the cards in the right order.
Once you've completed the task, can you recreate the method for your partner without looking at the cards?"

The interactive proof sorters which are available can be used as an alternative to the printable cards, or for students to check their suggested arrangements of the cards.

### Key questions

What is special about the new quadrilateral formed by joining the midpoints?
Does it always happen?
Does it help to draw in the diagonals?
Are there any similar shapes in your picture?
Does that help you to work out the relationship between the areas?

### Possible support

Students could start by exploring what happens when you join the midpoints of a triangle.