### 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

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

# Running Race

##### Stage: 4 Short Challenge Level:

In a leisure park there are three running tracks, all with the same Start and Finish, and all made from either one or two semicircles with centres on the same line.
Three runners $P$, $Q$ and $R$ start together at the Start and run at the same constant speed along the tracks shown. In what order do they finish?

##### Hint
Imagine a circle of diameter 1 unit, and another with the same diameter, and another.

What is the total perimeter of these three circles (you can leave $\pi$ in your answer)?

What is the total perimeter of a circle of diameter $3$?

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Now imagine two circles, one of diameter $2$ units and one of diameter $3$ units.

What is the total perimeter of these two circles?

What is the perimeter of a circle of diameter $5$ units?

What do you notce this time?

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Is this generally the case with combinations of circles whose total diameters are equal?

Can you explain why?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.
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