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Triangle Power


The total power of a triangle is defined by

 P=1/2(a_1^2+a_2^2+a_3^2),
(1)

where a_i are the side lengths, and the "partial power" is defined by

 p_1=1/2(a_2^2+a_3^2-a_1^2).
(2)

Then

 p_1=a_2a_3cosalpha_1
(3)
 P=p_1+p_2+p_3
(4)
 P^2+p_1^2+p_2^2+p_3^2=a_1^4+a_2^4+a_3^4
(5)
 Delta=1/2sqrt(p_2p_3+p_3p_1+p_1p_2)
(6)
 p_1=A_1H_2^_·A_1A_3^_
(7)
 (a_1p_1)/(cosalpha_1)=a_1a_2a_3=4DeltaR
(8)
 p_1tanalpha_1=p_2tanalpha_2=p_3tanalpha_3,
(9)

where Delta is the area of the triangle and H_i are the feet.

Finally, if a side of the triangle and the value of any partial power are given, then the locus of the third polygon vertex is a circle or straight line.


See also

Altitude, Circle Power, Perpendicular Foot, Triangle

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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.

Referenced on Wolfram|Alpha

Triangle Power

Cite this as:

Weisstein, Eric W. "Triangle Power." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrianglePower.html

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