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GOLDEN TRIANGLE

A golden rectangle is a triangle wherein the ratio of the its longer side to its shorter side is the golden ratio, which is 

f =  1.618...

Because of this golden ratio, it has been known to be the triangle of most aesthetic proportion.  The golden triangle forms the 5 points of a pentagram (or 5-point star).

The three angles of the golden triangles are 72, 36, and 36 degrees.  It can be noted that twice the cosine of 36 degrees is the golden ratio f.  The twice the cosine of 72 degrees is  1/f.

One can form smaller versions of golden triangle be drawing an bisector from one of the isoceles vertex to the opposite side.  This process can be done repeatedly.  A spiral running thru the vertices of these generated golden triangles form an equiangular spiral. 

A decagon is also related to the golden triangle.  A decagon is composed of ten golden triangle which shares a common vertex, lying at the center of the decagon.

A golden triangle can be drawn from a pentagon by connecting one side with the vertex directly opposite of the chosen side.  Five different golden triangles can be drawn from a pentagon.

 

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