Polar Circle
Given a Triangle, the polar circle has center at the Orthocenter 
. Call 
the Feet of theAltitude. Then the Radius is
 |  |  | (1) |
|  | (2) | |
|  | (3) |
where
is the Circumradius, 
the Vertex angles, and 
the correspondingside lengths.A Triangle is self-conjugate with respect to its polar circle. Also, the Radical Axis of any two polar circlesis the Altitude from the third Vertex. Any two polar circles of an OrthocentricSystem are orthogonal. The polar circles of the triangles of a Complete Quadrilateral constitute a Coaxal Systemconjugate to that of the circles on the diagonals.
See also Coaxal System, Orthocentric System, Polar, Pole, Radical Axis
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.
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