A golden triangle, also called a sublime triangle, is an isosceles triangle in which the duplicated side is in the golden ratio \varphi to the base side:

Angles

In other geometric figures

Logarithmic spiral

The golden triangle is used to form some points of a logarithmic spiral. By bisecting one of the base angles, a new point is created that in turn, makes another golden triangle. The bisection process can be continued indefinitely, creating an infinite number of golden triangles. A logarithmic spiral can be drawn through the vertices. This spiral is also known as an equiangular spiral, a term coined by René Descartes. "If a straight line is drawn from the pole to any point on the curve, it cuts the curve at precisely the same angle," hence equiangular. This spiral is different from the golden spiral: the golden spiral grows by a factor of the golden ratio in each quarter-turn, whereas the spiral through these golden triangles takes an angle of 108° to grow by the same factor.

Golden gnomon

Closely related to the golden triangle is the golden gnomon, which is the isosceles triangle in which the ratio of the equal side lengths to the base length is the reciprocal of the golden ratio \varphi. "The golden triangle has a ratio of base length to side length equal to the golden section φ, whereas the golden gnomon has the ratio of side length to base length equal to the golden section φ."

Angles

(The distances AX and CX are both a′ = a = φ, and the distance AC is b′ = φ², as seen in the figure.)

Bisections

Tilings