Sine-triple-angle circle

In triangle geometry, the sine-triple-angle circle is one of a circle of the triangle.^[1][2] Let A₁ and A₂ points on BC , a side of triangle ABC . And, define B₁, B₂, C₁ and C₂ similarly for CA and AB. If

Sine-Triple-Angle Circle

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

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

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] At first, Tucker and Neuberg called the circle "cercle triplicateur".^[4]

Properties

• ${\displaystyle |A_{1}A_{2}|:|B_{1}B_{2}|:|C_{1}C_{2}|=\sin 3A:\sin 3B:\sin 3C}$.^[5] 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(49) (see below for X₄₉).^[6]
• 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(184), the inverse of Jerabek center in Brocard circle, and X(1147).^[7]
• Intersections of Polar of A,B and C with the circle and BC,CA and AB are colinear.^[8]
• The radius of sine-triple-angle circle is

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

where R is the circumradius of triangle ABC.

Center

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

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

Generalization

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

Also,

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

See also

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