- Source: Sine-triple-angle circle
In triangle geometry, the sine-triple-angle circle is one of a circle of the triangle. 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
∠
A
=
∠
A
B
1
C
1
=
A
C
2
B
2
,
{\displaystyle \angle A=\angle AB_{1}C_{1}=AC_{2}B_{2},}
∠
B
=
∠
B
C
1
A
1
=
B
A
2
C
2
,
{\displaystyle \angle B=\angle BC_{1}A_{1}=BA_{2}C_{2},}
and
∠
C
=
∠
C
A
1
B
1
=
C
B
2
A
2
,
{\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. At first, Tucker and Neuberg called the circle "cercle triplicateur".
Properties
|
A
1
A
2
|
:
|
B
1
B
2
|
:
|
C
1
C
2
|
=
sin
3
A
:
sin
3
B
:
sin
3
C
{\displaystyle |A_{1}A_{2}|:|B_{1}B_{2}|:|C_{1}C_{2}|=\sin 3A:\sin 3B:\sin 3C}
. 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).
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).
Intersections of Polar of A,B and C with the circle and BC,CA and AB are colinear.
The radius of sine-triple-angle circle is
R
|
1
+
8
cos
(
A
)
cos
(
B
)
cos
(
C
)
|
,
{\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. The trilinear coordinates of X(49) is
cos
(
3
A
)
:
cos
(
3
B
)
:
cos
(
3
C
)
{\displaystyle \cos(3A):\cos(3B):\cos(3C)}
.
Generalization
For natural number n>0, if
∠
A
1
C
1
A
2
=
(
2
n
−
1
)
A
−
(
n
−
1
)
π
,
{\displaystyle \angle A_{1}C_{1}A_{2}=(2n-1)A-(n-1)\pi ,}
∠
B
1
A
1
B
2
=
(
2
n
−
1
)
B
−
(
n
−
1
)
π
,
{\displaystyle \angle B_{1}A_{1}B_{2}=(2n-1)B-(n-1)\pi ,}
and
∠
C
1
B
1
C
2
=
(
2
n
−
1
)
C
−
(
n
−
1
)
π
,
{\displaystyle \angle C_{1}B_{1}C_{2}=(2n-1)C-(n-1)\pi ,}
then A1, A2, B1, B2, C1 and C2 are concyclic. Sine-triple-angle circle is the special case in n=2.
Also,
|
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
{\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
References
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.
External links
Weisstein, Eric W. "Sine-Triple-Angle Circle". MathWorld.
GeoGebra,X(49) Center of sine-triple-angle circle