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?

Triangles Within Pentagons

Stage: 4 Challenge Level:

Curt from Reigate College started by getting a general formula for Triangle numbers

$T_n = 1 + 2 + 3+ . . . +(n-1) + n$

$T_n = n + (n-1) + . . . + 3+2+1$

The second line is just a repeat of the first but with the terms in reverse order.

Then adding the two lines the right hand side becomes $n$ lots of $n+1$

So $2T_n=n(n+1)$ and since everything was added in twice,

$T_n=\frac{n(n+1)}{2}$.

David from Sha Tin College used a known general result for any progression that steps up by a fixed amount term on term

$\frac{n}{2}(2a+(n-1)d)$

where n is the number of terms, a is the starting term and d is that step up.

If the step up is equal to 1, and the start value is also 1, you can see that this is the same result as Curt demonstrated.

Continuing with the Pentagonal numbers
The diagram shows $P_4=3T_3+4$

And in general $P_n=3T_{n-1}+n$

Now using the result for $T_n$ to see $P_n$ in terms of $n$

$P_n=\frac{3n(n-1)}{2}+n$

We now have to show that three times that result is a triangle number

$3P_n=3[\frac{3n(n-1)+2n}{2}]$ and is the same as $3[\frac{3n^2-n}{2}]$

or

$\frac{3n(3n-1)}{2}$

And that's the same as $T_{3n-1}$ the required result.