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.

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".

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] The trilinear coordinates of X(49) is

For natural number n>0, if

− ( n − 1 ) π ,

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

− ( n − 1 ) π ,

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

− ( 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.

[8] Sine-triple-angle circle is the special case in n=2.

= sin ⁡ ( 2 n − 1 )

: sin ⁡ ( 2 n − 1 )