Polar circle (geometry)

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orthocenter

H

; intersect extended sides

△ ABC

D, E, F

Polar circle of

△ ABC

H

In

and whose squared radius is

{\begin{aligned}r^{2}&={\overline {HA}}\times {\overline {HD}}={\overline {HB}}\times {\overline {HE}}={\overline {HC}}\times {\overline {HF}}\\[4pt]&=-4R^{2}\cos A\cos B\cos C\\[4pt]&=4R^{2}-{\frac {a^{2}+b^{2}+c^{2}}{2}}\end{aligned}}

where $A, B, C$ denote both the triangle's

circumradius (the radius of its circumscribed circle); and

a, b, c

are the lengths of the triangle's sides opposite vertices

A, B, C

respectively.^[1]

The first parts of the radius formula reflect the fact that the orthocenter divides the altitudes into segment pairs of equal products. The

Properties

Any two polar circles of two triangles in an

orthogonal.^[1]

The polar circles of the triangles of a

coaxal system.^[1]

Reference triangle $△ ABC$ and its tangential triangle

  Circumcircle of $△ ABC$
( $e$ ; centered at circumcenter $L$ )

  Circumcircle of tangential triangle
( $s$ ; centered at $K$ )

  Nine-point circle of $△ ABC$
( $t$ ; centered at nine-point center $M$ )

  Polar circle of $△ ABC$
( $d$ ; centered at orthocenter $H$ )
The centers of these circles relating to $△ ABC$ are all collinear–they fall on the Euler line.

A triangle's circumcircle, its nine-point circle, its polar circle, and the circumcircle of its tangential triangle are coaxal.^[2]^{: p. 241}

References

^ ^a ^b ^c Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
^ Altshiller-Court, Nathan, College Geometry, Dover Publications, 2007 (orig. 1952).

External links

Weisstein, Eric W. "Polar Circle". MathWorld.

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