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

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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?

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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?

Triangles Within Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Well explained by Tom, from Cottenham Village College

The answer I got for $T_n$ is: $8T_n+1=(2n+1)^2$

The $2n+1$ part is because the diagram looks like this for $T_3$
square arrangement
$2T_n$ form a rectangle $n$ by $n+1$

The four rectangles rotate around the centre and together make a square of side $n+(n+1)$

So we get the equation $8T_n+1=(2n+1)^2$