1041. Pythagorean triangles

    A Pythagorean triple  (a, b, c)  consists of three natural numbers  a < b < c  such that

a² + b² = c²

    A triangle is Pythagorean whether its three sides,  a, b, c,  form a Pythagorean triple, that is, a right angled triangle whose sides mesure natural numbers (the length unit doesn't matter). The most famous Pythagorean triangle is the Egyptian triangle, whose sides measure 3, 4 and 5. The Egyptian rope (with 12 knots) is a tool to make right angles...

    Pepe Chapuzas asked if there was a plane that formed 3 Pythagorean triangles with the planes x=0, y=0 and z=0. I proposed this question as a challenge. The next day Nina Guindilla said that she had found out one, but only gave as clues the hypotenuses of the three Pythagorean triangles formed:

    Pepe Chapuzas wrote this system of equations:

E₁ :    x² + y² = 267² = 71289

E₂ :    x² + z² = 125² = 15625

E₃ :    y² + z² = 244² = 59536

    And solved it:

E₁ + E₂ – E₃ :    2x² = 27378 ;   x = 117

E₁ + E₃ – E₂ :    2y² = 115200 ;   y = 240

E₂ + E₃ – E₁ :    2x² = 3872 ;   z = 44

    The plane was

x/117 + y/240 + z/44 = 1