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?

Tri-split

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

A Little Light Thinking

Stage: 4 Challenge Level:

This problem follows on from Charlie's Delightful Machine, so take a look at that first.

The rules for turning on the lights of Charlie's Delightful Machine are all given by linear sequences (like those found in Shifting Times Tables).

In the first problem, you found efficient strategies for working out the rules controlling each light.

Now try to make two lights light up at once.

Once you have made a pair of lights light up simultaneously, can you find another number that will light them both up? And another? And another? ...

Is there a connection between the rules that light up each individual light and the rule that lights up the pair?

Sometimes it's impossible to switch a pair of lights on simultaneously.
How can you decide whether it is possible to switch a pair of lights on simultaneously?

Now explore turning on three or even all four lights.

If you find an example where it's impossible to light them all up, try to explain why it's not possible.

If you find an example where it is possible to light them all up, work out what is special about the sequence of numbers that light up all four lights simultaneously.