Sine-triple-angle circle
In triangle geometry, the sine-triple-angle circle is one of a circle of the triangle. Let and points on, a side of triangle . And, define and similarly for and. If
and
then and lie on a circle called the sine-triple-angle circle. At first, Tucker and Neuberg called the circle "cercle triplicateur".
Properties
- . This property is the reason why the circle called "sine-triple-angle circle". But, the number of circle which cuts three sides of triangle that satisfies the ratio are countless. The centers of these circles are on the hyperbola through the incenter, three excenters, and X.
- The homothetic centers of Nine-point circle and the circle are the Kosnita point and the focus of Kiepert parabola.
- The homothetic centers of circumcircle and the circle are X, the inverse of Jerabek center in Brocard circle, and X.
- Intersections of Polar of and with the circle and and are colinear.
- The radius of sine-triple-angle circle is
Center
The center of sine-triple-angle circle is a triangle center designated as X in Encyclopedia of Triangle Centers. The trilinear coordinates of X isGeneralization
For natural number n>0, ifand
then and are concyclic. Sine-triple-angle circle is the special case in n=2.
Also,