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

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

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.