viernes, 27 de enero de 2017

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:

E:    x2 + y2 = 2672 = 71289
E:    x2 + z2 = 1252 = 15625
E:    y2 + z2 = 2442 = 59536
    And solved it:
E1 + E2 – E:    2x2 = 27378 ;   x = 117
E1 + E3 – E:    2y2 = 115200 ;   y = 240
E2 + E3 – E:    2x2 = 3872 ;   z = 44

    The plane was
x/117 + y/240 + z/44 = 1

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