geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the polar circle of a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

is the

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

whose center is the triangle's orthocenter and whose squared radius is :

\begin
r^2 & = HA\times HD=HB\times HE=HC\times HF \\
& =-4R^2\cos A \cos B \cos C=4R^2-\frac(a^2+b^2+c^2),
\end

where ''A, B, C'' denote both the triangle's vertices and the

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...

measures at those vertices, ''H'' is the orthocenter (the intersection of the triangle's altitudes), ''D'', ''E'', ''F'' are the feet of the altitudes from vertices ''A, B, C'' respectively, ''R'' is the triangle's

circumradius In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every pol ...

(the radius of its

circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

), and ''a'', ''b'', ''c'' are the lengths of the triangle's sides opposite vertices ''A'', ''B'', ''C'' 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.

Properties

Any two polar circles of two triangles in an

orthocentric system In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and ...

are

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

. The polar circles of the triangles of a

complete quadrilateral In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six ...

form a coaxal system. A triangle's circumcircle, its

nine-point circle In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of ea ...

, its polar circle, and the circumcircle of its tangential triangle 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