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# hard geometry

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We are given a triangle with a perimeter of 272 and the product of its sides equal to 314.  What is the product of the inradius and circumradius?

Jun 16, 2020

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We are given a triangle with a perimeter of 272 and the product of its sides equal to 314.

$$\text{Let inradius r = \dfrac{2A}{a+b+c}  } \\ \text{Let circumradius  R =\dfrac{abc}{4A}  } \\ a+b+c = 272 \\ abc = 314$$

$$\begin{array}{|rcll|} \hline rR &=& \dfrac{2A}{(a+b+c)} * \dfrac{abc}{4A} \\\\ rR &=& \dfrac{abc}{2(a+b+c)} \\\\ rR &=& \dfrac{314}{2(272)} \\\\ rR &=& \dfrac{157}{272} \\\\ \mathbf{rR} &=& \mathbf{0.57720588235} \\ \hline \end{array}$$

Jun 16, 2020