Editing Golden ratio (section)

=====Kepler triangle=====
{{main|Kepler triangle}}
{{multiple image
|image1=Kepler triangle.svg|caption1=Geometric progression of areas of squares on the sides of a Kepler triangle
|image2=Kepler and the Deathly Hallows.svg|caption2=An isosceles triangle formed from two Kepler triangles maximizes the ratio of its inradius to side length
|total_width=480}}
The ''Kepler triangle'', named after [[Johannes Kepler]], is the unique [[right triangle]] with sides in [[geometric progression]]:
<math display=block>
1\mathbin:\sqrt{\varphi\vphantom+}\mathbin:\varphi.</math>
These side lengths are the three [[Pythagorean mean]]s of the two numbers {{tmath|\varphi \pm 1}}. The three squares on its sides have areas in the golden geometric progression {{tmath|\textstyle 1\mathbin:\varphi\mathbin:\varphi^2}}.

Among isosceles triangles, the ratio of [[inradius]] to side length is maximized for the triangle formed by two [[Reflection (mathematics)|reflected copies]] of the Kepler triangle, sharing the longer of their two legs.<ref name="Liber mensurationum" /> The same isosceles triangle maximizes the ratio of the radius of a [[semicircle]] on its base to its [[perimeter]].<ref name=bruce />

For a Kepler triangle with smallest side length {{tmath|s}}, the [[area]] and [[acute angle|acute]] [[internal angle]]s are:
<math display=block>\begin{align}
A &= \tfrac12 s^2\sqrt{\varphi\vphantom+}, \\[5mu]
\theta &= \sin^{-1}\frac{1}{\varphi}\approx 38.1727^\circ\!, \\[5mu]
\theta &= \cos^{-1}\frac{1}{\varphi}\approx 51.8273^\circ\!.
\end{align}</math>