Law of Sines - Relation To The Circumcircle

Relation To The Circumcircle

In the identity

the common value of the three fractions is actually the diameter of the triangle's circumcircle. It can be shown that this quantity is equal to

\begin{align} \frac{abc} {2S} & {} = \frac{abc} {2\sqrt{s(s-a)(s-b)(s-c)}} \\ & {} = \frac {2abc} {\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4) }}, \end{align}

where S is the area of the triangle and s is the semiperimeter

The second equality above is essentially Heron's formula.

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