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 ABC, let A₁ and A₂ be points on side BC , with B₁, B₂, C₁ and C₂ defined similarly on CA and AB respectively. If

${\displaystyle \angle A=\angle AB_{1}C_{1}=AC_{2}B_{2},}$

${\displaystyle \angle B=\angle BC_{1}A_{1}=BA_{2}C_{2},}$

and

${\displaystyle \angle C=\angle CA_{1}B_{1}=CB_{2}A_{2},}$

then A₁, A₂, B₁, B₂, C₁ and C₂ lie on a circle called the sine-triple-angle circle,^[3] originally referred to by Tucker and Neuberg as the cercle triplicateur.^[4]

• ${\displaystyle |A_{1}A_{2}|:|B_{1}B_{2}|:|C_{1}C_{2}|=\sin 3A:\sin 3B:\sin 3C}$.^[5], 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 X₄₉).^[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 A,B and C with the circle and BC,CA and AB are colinear.^[8]
• The radius of the sine-triple-angle circle is

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

where R is the circumradius of triangle ABC.

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

${\displaystyle \cos(3A):\cos(3B):\cos(3C)}$.

For a given natural number n>0, if

${\displaystyle \angle A_{1}C_{1}A_{2}=(2n-1)A-(n-1)\pi ,}$

${\displaystyle \angle B_{1}A_{1}B_{2}=(2n-1)B-(n-1)\pi ,}$

and

${\displaystyle \angle C_{1}B_{1}C_{2}=(2n-1)C-(n-1)\pi ,}$

then

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

and

A₁, A₂, B₁, B₂, C₁ and C₂ are concyclic.^[8] The sine-triple-angle circle is the special case where n=2.

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

• V. Thebault (1965). Sine-triple-angle-circle. Vol. 65. Mathesis. pp. 282–284.
• Ehrmann, Jean-Pierre; Lamoen, Floor van (2002). The Stammler Circles. Forum Geometricorum. pp. 151–161.

• Weisstein, Eric W. "Sine-Triple-Angle Circle". MathWorld.
• GeoGebra, X(49) Center of sine-triple-angle circle