Sublime Triangle

(Also called the Golden Triangle)

The Sublime Triangle is an isosceles triangle with angles of measure 36, 72, and 72 degrees. It is the only triangle with angle measures in the ratio 1:2:2.

Prove that the ratio of a lateral side of a sublime triangle to its base is the golden ratio, , because

Alternatively, if we DEFINE the Sublime triangle as an isosceles triangle with the ratio of the measure of the lateral side to the measure of the base to be the Golden Ratio, then we can prove that the angles have measure of 36 - 72 - 72.

Hint: Construct an angle bisector of one of the base angles and extend it to the opposite side, thus producing a smaller triangle similar to the original.

Prove that

Hint: Help with a notation?


The Sublime Triangle embedded in other figures (prove each of these cases):

In a regular PentagonFind others?

In a regular DecagonFind Others?

In a Pentagram Find others?

Find the Sublime triangle embedded in other figures?

Look for other "embedded" sublime triangles by extending sides or diagonals of a regular polygon.

See also Closed Form of Some Cosine and Sine Values

See also Cos 36 degrees