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Power (Triangle)

The total Power of a Triangle is defined by

\begin{displaymath}
P\equiv {\textstyle{1\over 2}}({a_1}^2+{a_2}^2+{a_3}^2),
\end{displaymath} (1)

where $a_i$ are the side lengths, and the ``partial power'' is defined by
\begin{displaymath}
p_1\equiv {\textstyle{1\over 2}}({a_2}^2+{a_3}^2-{a_1}^2).
\end{displaymath} (2)

Then
\begin{displaymath}
p_1=a_2a_3\cos\alpha_1
\end{displaymath} (3)


\begin{displaymath}
P=p_1+p_2+p_3
\end{displaymath} (4)


\begin{displaymath}
P^2+{p_1}^2+{p_2}^2+{p_3}^2={a_1}^4+{a_2}^4+{a_3}^4
\end{displaymath} (5)


\begin{displaymath}
\Delta={\textstyle{1\over 2}}\sqrt{p_2p_3+p_3p_1+p_1p_2}
\end{displaymath} (6)


\begin{displaymath}
p_1=\overline{A_1H_2}\cdot\overline{A_1A_3}
\end{displaymath} (7)


\begin{displaymath}
{a_1p_1\over\cos\alpha_1}=a_1a_2a_3=4\Delta R
\end{displaymath} (8)


\begin{displaymath}
p_1\tan\alpha_1=p_2\tan\alpha_2=p_3\tan\alpha_3,
\end{displaymath} (9)

where $\Delta$ is the Area of the Triangle and $H_i$ are the Feet of the Altitudes. Finally, if a side of the Triangle and the value of any partial power are given, then the Locus of the third Vertex is a Circle or straight line.

See also Altitude, Foot, Triangle


References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 260-261, 1929.




© 1996-9 Eric W. Weisstein
1999-05-26