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.

[1] It is an isosceles triangle which is obtuse with an irrational but algebraic ratio between the lengths of its sides and its base.

[2] 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.

The triangle △ABC is isosceles which has the same length of sides as AB = AC.

If the ratio of the base to either leg is x, we can set that AB = AC = 1, BC = x.

Let H be the foot of the perpendicular drawn from the apex A to the base.

Let □IJKA be a square on base AC with its side length as b.

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

and Then f is monotonically increasing function and by Intermediate value theorem, the Calabi's equation f(x) = 0 has unique solution in open interval

The value of x is calculated by Newton's method as follows: The value of x can expressed with complex numbers by using Cardano's method: The value of x can also be expressed without complex numbers by using Viète's method: The value of x 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, ...] = The Calabi triangle is obtuse with base angle θ and apex angle ψ as follows: The rational approxmation of x is ⁠h95/k95⁠ and an error bounds ε is as follows:

Example figure of Calabi triangle 01
Example figure of Calabi triangle 01
Example figure of Calabi triangle 02
Example figure of Calabi triangle 02
Example figure of Calabi triangle 03
Example figure of Calabi triangle 03
Example figure of Calabi triangle 04
Example figure of Calabi triangle 04