Given an obtuse triangle, the polar circle has center at the orthocenter 
. Call 
the feet. Then the
square of the radius 
is given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
where 
is the circumradius, 
, 
,
and 
are the angles, and 
, 
,
and 
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
 |
(6)
|
and trilinear equation
 |
(7)
|
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.
