Why do this problem?
This problem leads to a result which is easy to guess
visually but not so easy to prove.
Geometrical, numerical, and algebraic ideas can all be used
to reach a solution, and properties of averaging can also come
out of the problem. Numerical patterns can be investigated using
a spreadsheet.
Possible approach
Use the interactivity (or accurately construct some triangles
with their inscribed circles) and see what happens to the
angles in the nested triangles.In order to see why this is
happening, it's important to make sure everyone knows that the
centre of the inscribed circle is at the point where the angle
bisectors of the original triangle meet, and that radii meet
tangents at a right angle. This information can be used to
write expressions for the three angles in the new triangle in
terms of the original angles.
Students could now create a spreadsheet which allows them to
input three angles which sum to 180 degrees and use their
expressions to work out the three new angles. By continuing the
sequence, the angles quickly converge.
Key questions
What seems to be happening to the angles in each new triangle
that we draw?
How can we calculate the angles of each new triangle if we know
the original angles?
Possible extension
Investigate the sequence $(90-x/2), 90-(90-x/2)/2$ and so on to
explain why the angles converge to their limit.
The three new angles are each the mean of a pair of the
original angles. In general, what happens if you keep finding
the mean of pairs of numbers to give three new numbers?
Possible support
Work with numerical examples and try to
explain the patterns formed.