Polar circle (geometry)

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

The trigonometric formula for the radius shows that the polar circle has a real existence only if the triangle is obtuse, so one of its angles is obtuse and hence has a negative cosine.

Any two polar circles of two triangles in an orthocentric system are orthogonal.[1]: p.

177 The polar circles of the triangles of a complete quadrilateral form a coaxal system.[1]: p.

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

Reference triangle △ ABC
Altitudes (concur at orthocenter H ; intersect extended sides of △ ABC at D, E, F )
Polar circle of △ ABC , centered at H
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 .