Calabi triangle
The Calabi triangle is a special triangle found by Eugenio Calabi and defined by its property of having three different placements for the largest square that it contains. It is an isosceles triangle which is obtuse with an irrational but algebraic ratio between the lengths of its sides and its base.

Definition
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. If the largest such square can be positioned in three different ways, then the triangle is either an equilateral triangle or the Calabi triangle. Thus, the Calabi triangle may be defined as a triangle that is not equilateral and has three placements for its largest square.

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
The condition is .
In this case  is valid for equilateral triangle.
case 2)  is right triangle
The condition is .
In this case no value is vaild.
case 3)  is obtuse triangle
The condition is .
In this case the Calabi triangle is valid for the value  of .

Consider the case of . Then

Let a base angle be  and a square be  on base  with it's 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 it's 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 it's 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:

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:[1, 1, 1, 4, 2, 1, 2, 1, 5, 2, 1, 3, 1, 1, 390, ...] =
.

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

See also 

Biggest little polygon
Cubic equation

Footnotes

Notes

Citations

References 
 - Galois Theory Errata.

External links 

Types of triangles