Sine-triple-angle circle

In triangle geometry, the sine-triple-angle circle is one of many circles that can be defined from a triangle. For triangle , let and be points on side , with and defined similarly on and respectively. If

and

then and lie on a circle called the sine-triple-angle circle, originally referred to by Tucker and Neuberg as the cercle triplicateur.

Properties

• ., which gives the circle it's name. However, there are an uncountably infinite amount of circles that also satisfy this identity. The centers of these circles are on the hyperbola through the incenter, three excenters, and X(49) (see below for X).
• The homothetic centers of the Nine-point circle and the sine-triple-angle circle is the Kosnita point and the focus of the Kiepert parabola.
• The homothetic centers of the circumcircle and the sine-triple-angle circle is X(184), the inverse of Jerabek center in Brocard circle, and X(1147).
• Intersections of the Polar of and with the circle and and are colinear.
• The radius of the sine-triple-angle circle is

where is the circumradius of triangle .

Center

The center of sine-triple-angle circle is a triangle center designated as X(49) in Encyclopedia of Triangle Centers. with trilinear coordinates

.

Generalization

For a given natural number n>0, if

and

then

and

and are concyclic. The sine-triple-angle circle is the special case where n=2.

See also

• Taylor circle
• Tucker circle
• Triangle conic
• Triple angle

References

External links

• GeoGebra, X(49) Center of sine-triple-angle circle