Foster, Colin (2016) Proof without words: integer right
triangle hypotenuses without Pythagoras. College
Mathematics Journal, 47 (2). p. 101. ISSN 1931-1346
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Proof Without Words: Integer Right Triangle Hypotenuses Without
Pythagoras
Colin Foster ([email protected]), University of Nottingham, United
Kingdom
Theorem. A right triangle with legs 3 and 4 has hypotenuse 5.
Proof.
A referee offered the following generalization. Consider integers m > n > 0.
Draw line segments from (0, 0) to (2mn, 2n 2 ) and from (0, m 2 + n 2 ) to (mn, n 2 );
these are perpendicular. The corresponding right triangle has legs m 2 − n 2 and 2mn
and hypotenuse m 2 + n 2 . Any primitive right triangle is given by such a pair with
the additional criteria that m and n are relatively prime and have opposite parity [1].
Summary. Without reference to the Pythagorean theorem, we show that a right triangle
with legs 3 and 4 has hypotenuse 5. The figure can be modified for other integer right
triangles.
Reference
1. W. Sierpiński, Pythagorean Triangles. Trans. A. Sharma. Yeshiva Univ., New York, 1962. Reprinted
by Dover, Minneola, NY, 2003.
http://dx.doi.org/10.4169/college.math.j.47.2.101
MSC: 97G40
VOL. 47, NO. 2, MARCH 2016 THE COLLEGE MATHEMATICS JOURNAL
101