Base triple AC BC CD AB 20 = 4×5 15 = 3×5 12 = 3×4 25 = 52 156 = 12×13 65 = 5×13 60 = 5×12 169 = 132 255 = 15×17 136 = 8×17 120 = 8×15 289 = 172 600 = 24×25 175 = 7×25 168 = 7×24 625 = 252 609 = 21×29 580 = 20×29 420 = 20×21 841 = 292All positive integer primitive inverse-Pythagorean triples having up to three digits, with the hypotenuse for comparison
In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theoremR. B. Nelsen, Proof Without Words: A Reciprocal Pythagorean Theorem, Mathematics Magazine, 82, December 2009, p. 370 or the upside down Pythagorean theoremThe upside-down Pythagorean theorem, Jennifer Richinick, The Mathematical Gazette, Vol. 92, No. 524 (July 2008), pp. 313-316) is as follows:Johan Wästlund, "Summing inverse squares by euclidean geometry", http://www.math.chalmers.se/~wastlund/Cosmic.pdf, pp. 4–5.
Let , be the endpoints of the hypotenuse of a right triangle . Let be the foot of a perpendicular dropped from , the vertex of the right angle, to the hypotenuse. Then
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.
Proof The area of triangle can be expressed in terms of either and , or and :
given , and .
Using the Pythagorean theorem,
as above.
Note in particular:
Special case of the cruciform curve The cruciform curve or cross curve is a quartic plane curve given by the equation
where the two parameters determining the shape of the curve, and are each .
Substituting with and with gives
Inverse-Pythagorean triples can be generated using integer parameters and as follows.
Application
If two identical lamps are placed at and , the theorem and the inverse-square law imply that the light intensity at is the same as when a single lamp is placed at .
See also
References