Golden triangle (mathematics)
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A golden triangle is a triangle in which two of the sides have equal length and in which the ratio of this length to that of the third, smaller side is the golden ratio. These triangles are isosceles and are commonly found in the points of pentagrams.
The golden triangle is an isosceles triangle such that the ratio of the hypotenuse a to base b is equal to the golden ratio, a/b==phi. From the above figure, this means that the triangle has vertex angle equal to theta==2sin^(-1)(b/(2a))==2sin^(-1)(1/(2phi))==1/5pi, (1)
or 36 degrees. GoldenTriangleFigures
The triangles at the tips of a pentagram (left figure) and obtained by dividing a decagon by connecting opposite vertices (right figure) are golden triangles. This follows from the fact that a/b==phi (2)
for a pentagram and that the circumradius R of a decagon of side length s is R==phis. (3)
Golden triangles and gnomons can be dissected into smaller triangles that are golden gnomons and golden triangles (Livio 2002, p. 79).
Successive points dividing a golden triangle into golden gnomons and triangles lie on a logarithmic spiral (Livio 2002, p. 119).
Kimberling (1991) defines a second type of golden triangle in which the ratio of angles is phi:1, where phi is the golden ratio.
[edit] External links
Wolfram MathWorld's golden triangle article
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