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SSS Theorem


Specifying three sides uniquely determines a Triangle whose Area is given by Heron's Formula,

A = \sqrt{s(s-a)(s-b)(s-c)},
\end{displaymath} (1)

s\equiv {\textstyle{1\over 2}}(a+b+c)
\end{displaymath} (2)

is the Semiperimeter of the Triangle. Let $R$ be the Circumradius, then
A={abc\over 4R}.
\end{displaymath} (3)

Using the Law of Cosines
$\displaystyle a^2$ $\textstyle =$ $\displaystyle b^2+c^2-2bc\cos A$ (4)
$\displaystyle b^2$ $\textstyle =$ $\displaystyle a^2+c^2-2ac\cos B$ (5)
$\displaystyle c^2$ $\textstyle =$ $\displaystyle a^2+b^2-2ab\cos C$ (6)

gives the three Angles as
$\displaystyle A$ $\textstyle =$ $\displaystyle \cos^{-1}\left({b^2+c^2-a^2\over 2bc}\right)$ (7)
$\displaystyle B$ $\textstyle =$ $\displaystyle \cos^{-1}\left({a^2+c^2-b^2\over 2ac}\right)$ (8)
$\displaystyle C$ $\textstyle =$ $\displaystyle \cos^{-1}\left({a^2+b^2-c^2\over 2ab}\right).$ (9)

See also AAA Theorem, AAS Theorem, ASA Theorem, ASS Theorem, Heron's Formula, SAS Theorem, Semiperimeter, Triangle

© 1996-9 Eric W. Weisstein