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## 'Triangle Incircle Iteration' printed from http://nrich.maths.org/

Start with any triangle. Draw its inscribed circle
(the circle which just touches each side of the triangle). Draw the
triangle which has its vertices at the points of contact between
your original triangle and its incircle. Now keep repeating this
process starting with the new triangle 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...)