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Polar Circle


PolarCircle

Given an obtuse triangle, the polar circle has center at the orthocenter H. Call H_i the feet. Then the square of the radius r is given by

r^2=HA^_·HH_A^_
(1)
=HB^_·HH_B^_
(2)
=HC^_·HH_C^_
(3)
=-4R^2cosAcosBcosC
(4)
=4R^2-1/2(a^2+b^2+c^2),
(5)

where R is the circumradius, A, B, and C are the angles, and a, b, and c are the corresponding side lengths.

It is the anticomplement of the de Longchamps circle.

The polar circle, when it is defined, therefore has circle function

 l=-cosA
(6)

and trilinear equation

 (aalpha+bbeta+cgamma)(alphacosA+betacosB+gammacosC) 
 +calphabeta+balphagamma+abetagamma=0.
(7)
PolarCircleOrthogonal

It is orthogonal to the orthoptic circle of the Steiner inellipse, second Droz-Farny circle, and Stevanović circle.

A triangle is self-conjugate with respect to its polar circle. Also, the radical line of any two polar circles is the altitude from the third polygon vertex. Any two polar circles of an orthocentric system are orthogonal. The polar circles of the triangles of a complete quadrilateral constitute a coaxal system conjugate to that of the circles on the diagonals.

The polar triangle of the polar circle is the reference triangle.


See also

Coaxal System, Inversion Pole, Orthocentric System, Polar, Radical Line

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References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 136-138, 1967.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 176-181, 1929.

Referenced on Wolfram|Alpha

Polar Circle

Cite this as:

Weisstein, Eric W. "Polar Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolarCircle.html

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