Copyright © University of Cambridge. All rights reserved.
'Triangle Incircle Iteration' printed from https://nrich.maths.org/
Meg offers this solution:
The tangent of a circle is at right-angles to the radius of the
circle. That is, if you join the centre point of the circle to a
point where the circle meets the outer triangle, it makes an angle
of $ 90 ^{\circ} $ with the side of the triangle.
The bisector of the angles of triangle $ABC$ will all pass through
the centre of the circle.
From this we know that
$OAZ = \frac {a}{2}$ and $OZA = 90 ^{\circ}$
Hence $AOZ = 90- \frac{a}{2}$
Now consider triangle XOZ. This triangle is isosceles, so $OXZ =
XZO = \frac{180-(180-a)}{2} = \frac{a}{2}$
By similar arguments
$OXY = OYZ = \frac{b}{2}$ and $OYZ = OZY = \frac{c}{2}$ Hence the
new angles of the triangle are
$ZXY= \frac{a}{2}+\frac{b}{2}$
$XYZ = \frac{b}{2} + \frac{c}{2}$
$YZX =\frac{c}{2} + \frac{a}{2} $
$a+b+c=180 $
Hence $ \frac{a+b}{2}= 90 - \frac{c}{2}$
It follows that $ZXY =90 - \frac {c}{2}$.
A similar argument can be followed for $XYZ$ and $YZX$. If you
continue drawing triangles within circles, the angles will decrease
as shown here:
Triangle 1: a
Triangle 2: $ 90 - \frac{a}{2}$
Triangle 3: $90- \frac{90-\frac{a}{2}}{2}$ =$90 - \frac{90}{2} +
\frac{a}{4}$
Triangle 4: $90- \frac{90-\frac{90}{2}+\frac{a}{4}}{2}=
\frac{3}{4}.90 - \frac{a}{8}$
When you continue this iteration, you can demonstrate that the $a$
term becomes less and less significant, and the sum of the rest of
the terms tends to 60 degrees. Hence the triangle tends to an
equilateral triangle.