Polar circle (geometry)
In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is
where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter, D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius, and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.
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.Properties
Any two polar circles of two triangles in an orthocentric system are orthogonal.
The polar circles of the triangles of a complete quadrilateral form a coaxal system.
A triangle's circumcircle, its nine-point circle, its polar circle, and the circumcircle of its tangential triangle are coaxal.
