In triangle geometry, the sine-triple-angle circle is one of a circle of the triangle. Let "A"1 and "A"2 points on BC , a side of triangle ABC . And, define "B"1, "B"2, "C"1 and "C"2 similarly for CA and AB. If
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and
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then "A"1, "A"2, "B"1, "B"2, "C"1 and "C"2 lie on a circle called the sine-triple-angle circle. At first, Tucker and Neuberg called the circle "cercle triplicateur".
Properties.
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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
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Generalization.
For natural number n ;0, if
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and
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then "A"1, "A"2, "B"1, "B"2, "C"1 and "C"2 are concyclic. Sine-triple-angle circle is the special case in n=2.
Also,
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References.
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