Take any triangle whose sides have length \(a, b, c\). Find the largest area of a circle that fits inside the triangle. Write down a formula for this area in terms of \(a, b, c\). Can you find its radius?

In the case of an isosceles triangle of sides \(a, a, b\), it only takes an easy calculation to show that the largest embedded circular area is given by

\(\Large{ A = \frac14 \pi b^2 \left(\frac{2a-b}{2a+b}\right) }\).

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~(draw)~ and ~(roll)~ and ~(flip)~ work... try them out :P