Sine-triple-angle circle

In triangle geometry, the sine-triple-angle circle is one of many circles that can be defined from a triangle.^[1]^[2] For triangle Template:Mvar, let Template:Math and Template:Math be points on side Template:Mvar , with Template:Math and Template:Math defined similarly on Template:Mvar and Template:Mvar respectively. If

$∠ A = ∠ A B_{1} C_{1} = A C_{2} B_{2},$

$∠ B = ∠ B C_{1} A_{1} = B A_{2} C_{2},$

and

$∠ C = ∠ C A_{1} B_{1} = C B_{2} A_{2},$

then Template:Math and Template:Math lie on a circle called the sine-triple-angle circle,^[3] originally referred to by Tucker and Neuberg as the cercle triplicateur.^[4]

Properties

$| A_{1} A_{2} | : | B_{1} B_{2} | : | C_{1} C_{2} | = \sin 3 A : \sin 3 B : \sin 3 C$ .^[5], which gives the circle its 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 XTemplate:Sub).^[6]
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).^[7]
Intersections of the Polar of Template:Mvar and Template:Mvar with the circle and Template:Mvar and Template:Mvar are colinear.^[8]
The radius of the sine-triple-angle circle is

$\frac{R}{| 1 + 8 \cos (A) \cos (B) \cos (C) |},$

The center of sine-triple-angle circle is a triangle center designated as X(49) in Encyclopedia of Triangle Centers.^[7]^[9] with trilinear coordinates

$\cos (3 A) : \cos (3 B) : \cos (3 C)$ .

For a given natural number n>0, if

$∠ A_{1} C_{1} A_{2} = (2 n - 1) A - (n - 1) π,$

$∠ B_{1} A_{1} B_{2} = (2 n - 1) B - (n - 1) π,$

and

$∠ C_{1} B_{1} C_{2} = (2 n - 1) C - (n - 1) π,$

then

$| A_{1} A_{2} | : | B_{1} B_{2} | : | C_{1} C_{2} | = \sin (2 n - 1) A : \sin (2 n - 1) B : \sin (2 n - 1) C$

and

Template:Math and Template:Math are concyclic.^[8] The sine-triple-angle circle is the special case where n=2.