Kepler triangle
A Kepler triangle is a special right triangle with edge lengths in geometric progression. The progression can be written: 
or approximately 
When an isosceles triangle is formed from two Kepler triangles, reflected across their long sides, it has the maximum possible inradius among all isosceles triangles having legs of a given size. Most of the properties described below were discovered by the famous Russian mathematician Lev Emelianov.
Sides and angles of doubled Kepler triangle
Let’s define the doubled Kepler triangle as triangle which has the maximum possible inradius among all isosceles triangles having legs of a given size.
Let the incircle of an isosceles 
touch the sides 
and 
at points 
and 
![\[\angle BAM = \alpha, \angle ABC = 2 \beta, \alpha + 2 \beta = 90^\circ.\]](http://latex.artofproblemsolving.com/f/3/a/f3a138b76a9dfa594ca0abd5e3ee7a2fd562bfca.png)
We need to find minimum of
![\[\frac {AB}{r} = \cot \alpha +\cot \beta.\]](http://latex.artofproblemsolving.com/e/4/e/e4e973927aeeac6208840d5af2b4e6e05d6f52ef.png)
Let us differentiate this function with respect 
to taking into account that
![\[0<\alpha,2 \beta < 90^\circ, \frac {d \alpha}{d \beta} = -2: \frac {-2}{\sin^2 \alpha}+\frac {1}{\sin^2 \beta} = 0 \implies\]](http://latex.artofproblemsolving.com/2/f/7/2f7342b3433844456b72a5ed57a93b3b4c0c02e7.png)
![\[\sqrt{2} \sin \beta = \sin \alpha = \cos 2 \beta = 1 - 2 \sin^2 \beta \implies \sin \alpha = 1 - \sin^2 \alpha.\]](http://latex.artofproblemsolving.com/0/5/4/054c27edc3ca796892778c70a67c469d25861cf4.png)
Therefore 
Let 
![\[AI = \phi \cdot \sqrt{\phi}, BI = \sqrt{2} AI, IK = IM = \phi^2 \cdot \sqrt{\phi}.\]](http://latex.artofproblemsolving.com/8/d/0/8d0c39b1ebdc1f315b49b7836fdb20dfa967c91a.png)
vladimir.shelomovskii@gmail.com, vvsss
Construction of a Kepler triangle
Let 
be the midpoint of the base 
Point 
The point 
is the intersection of a circle with diameter 
and a circle centered at point 
and radius 
, which is located in the half-plane where there is no point 
.
The bisector of the obtuse angle between lines 
and intersects bisector at the vertex 
of the golden triangle.
The construction is based on the fact that 
