Largest circle in a triangle

MediumBrainteasers~5m

What is the area of the largest circle that fits inside a triangle with side lengths $16$ , $16$ , and $24$ ? The area can be written as $\dfrac{p\pi}{q}$ with $\dfrac{p}{q}$ in lowest terms. What is $p + q$ ?

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The largest circle inside a triangle is its incircle, with radius $r = (\text{area})/(\text{semiperimeter})$ .
Get the area from Heron's formula (or the isosceles height), then compute $\pi r^2$ .

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Largest circle in a triangle

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