Calabi triangle

The Calabi triangle is a special triangle found by Eugenio Calabi.
It is the unique triangle that has 3 different placements for the largest square that it contains, and is not the equilateral triangle. It is an isosceles triangle which is obtuse with an irrational but algebraic ratio between the lengths of its sides and its base.
Consider the largest square that can be placed in an arbitrary triangle. It may be that such a square could be positioned in the triangle in more than one way. In the equilateral triangle, the largest such square can be positioned in three different ways. Calabi found that there is exactly one other case, and so it is named the Calabi triangle.

Shape

The triangle is isosceles which has the same length of sides as. If the ratio of the base to either leg is, we can set that. Then we can consider the following three cases:
;case 1) is acute triangle:
;case 2) is right triangle:
;case 3) is obtuse triangle:
Consider the case of. Then
Let a base angle be and a square be on base with its side length as.
Let be the foot of the perpendicular drawn from the apex to the base. Then
Then and, so.
From △DEB ∽ △AHB,

case 1) is acute triangle

Let be a square on side with its side length as.
From △ABC ∽ △IBJ,
From △JKC ∽ △AHC,
Then
Therefore, if two squares are congruent,
In this case,
Therefore, it means that is equilateral triangle.

case 2) is right triangle

In this case,, so
Then no value is valid.

case 3) is obtuse triangle

Let be a square on base with its side length as.
From △AHC ∽ △JKC,
Therefore, if two squares are congruent,
In this case,
So, we can input the value of,
In this case,, we can get the following equation:

Root of Calabi's equation

If is the largest positive root of Calabi's equation:
we can calculate the value of by following methods.

Newton's method

We can set the function as follows:
The function is continuous and differentiable on and
Then is monotonically increasing function and by Intermediate value theorem, the Calabi's equation has unique solution in open interval.
The value of is calculated by Newton's method as follows:

NO	itaration value
	1.41421356237309504880168872420969807856967187537694...
	1.58943369375323596617308283187888791370090306159374...
	1.55324943049375428807267665439782489231871295592784...
	1.55139234383942912142613029570413117306471589987689...
	1.55138752458074244056538641010106649611908076010328...
	1.55138752454832039226341994813293555945836732015691...
	1.55138752454832039226195251026462381516359470986821...
	1.55138752454832039226195251026462381516359170380388...

Cardano's method

The value of can expressed with complex numbers by using Cardano's method:

Viète's method

The value of can also be expressed without complex numbers by using Viète's method:

Lagrange's method

The value of has continued fraction representation by Lagrange's method as follows:
=

Base angle and Apex angle

The Calabi triangle is obtuse with base angle and apex angle as follows: