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 Triangles

Stage: 4 Challenge Level:

This diagram shows how the first triangular number can be added to 3 copies of the second triangular number to make the fourth triangular number:

That is: $$T_1 + 3 \times T_2 = T_4$$ Here is a diagram showing how the second and third triangular numbers can be combined to make the sixth triangular number:

$$T_2 + 3 \times T_3 = T_6$$ Can you generalise this rule?

Can you find a rule in terms of $T_n$ and $T_{n+1}$?