geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the polar circle of a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

is the

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

whose center is the triangle's orthocenter and whose squared radius is where denote both the triangle's vertices and the

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

measures at those vertices; is the orthocenter (the intersection of the triangle's altitudes); are the feet of the altitudes from vertices respectively; is the triangle's circumradius (the radius of its circumscribed circle); and are the lengths of the triangle's sides opposite vertices respectively.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publications, 2007 (orig. 1960). 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 In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...

Properties

Any two polar circles of two triangles in an orthocentric system are

orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...

. The polar circles of the triangles of a complete quadrilateral form a coaxal system. The most important property of the polar circle is the triangle is self-polar; the polar of each side/point is the opposite side/point. A triangle's circumcircle, its nine-point circle, its polar circle, and the circumcircle of its

tangential triangle In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertex (geometry), vertices. Thus ...

are coaxal. Altshiller-Court, Nathan, ''College Geometry'', Dover Publications, 2007 (orig. 1952).

References

External links

* {{MathWorld, title=Polar Circle, urlname=PolarCircle Circles defined for a triangle