Inscribe two triangles 
and 
in a reference
triangle 
such that
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
Then the triangles 
and 
are both inscribed in a circle known as the sine-triple-angle
circle.
This circle is a central circle having circle function
 |
(4)
|
The center has triangle center function
 |
(5)
|
(Kimberling 1998, p. 74), which is Kimberling center 
,
and the circumradius is
 |
(6)
|
where 
is the circumradius of 
(Tucker and Neuberg 1887; Thébault 1956; Kimberling
1998, p. 234; typo corrected).
Te sine-triple-angle circle passes through Kimberling centers 
for 
, 3044, 3045, 3046, 3047, and 3048.
The distances satisfy the relationship
 |
(7)
|
(Thébault 1956), which gives the circle its name. The sine-triple-angle circle was originally called the cercle triplicateur by Tucker and Neuberg (1887).
In fact there are infinitely many circles that cut of the side line chords in the same proportions. The centers of these circles lie on the equilateral hyperbola through
the in- and excenters and 
(Ehrmann and van Lamoen 2002).
The similitude centers of the nine-point circle and the sine-triple-angle circle are the Kosnita point and the focus of the Kiepert parabola.
