Start with any triangle T ₁ and its inscribed circle. Draw the triangle T ₂ which has its vertices at the points of contact between the triangle T ₁ and its incircle. Now keep repeating this process starting with T ₂ to form a sequence of nested triangles and circles. What happens to the triangles?

If you wish, you can investigate this interactively:

If the angles in the first triangle are a, b and c prove that the angles in the second triangle are given (in degrees) by

f(x) = (90 - x/2)

where x takes the values a, b and c. Choose some triangles, investigate this iteration numerically and try to give reasons for what happens.

Investigate what happens if you reverse this process (triangle to circumcircle to triangle...)