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This theorem should not be confused with proposition 48 in book 1 of Euclid's Elements, the converse of the Pythagorean theorem, which states that if the square on one side of a triangle is equal to the sum of the squares on the other two sides then the other two sides contain a right angle.
For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio.
The only primitive Pythagorean triangles for which the square of the perimeter equals an integer multiple of the area are (3, 4, 5) with perimeter 12 and area 6 and with the ratio of perimeter squared to area being 24; (5, 12, 13) with perimeter 30 and area 30 and with the ratio of perimeter squared to area being 30; and (9, 40, 41) with ...
A Pythagorean triple is a set of three positive integers a, b, and c having the property that they can be respectively the two legs and the hypotenuse of a right triangle, thus satisfying the equation + =; the triple is said to be primitive if and only if the greatest common divisor of a, b, and c is one.
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Equivalently, by the Pythagorean theorem, they are the odd prime numbers for which is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime numbers for which itself is the hypotenuse of a primitive Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; is the hypotenuse of a right ...
Tesla and X CEO Elon Musk spent over a quarter of a billion dollars to help get President-elect Donald Trump back in the White House, according to newly released campaign finance records. The ...
The euro was down 0.13% at $1.0552 after rising 0.7% in the previous session in the wake of European Central Bank board member Isabel Schnabel saying that rate cuts should be gradual and move to ...