Inverse Pythagorean theorem

Base Pytha- gorean triple	AC	BC	CD	AB
(3, 4, 5)	20 = 4× 5	15 = 3× 5	12 = 3× 4	25 = 5²
(5, 12, 13)	156 = 12×13	65 = 5×13	60 = 5×12	169 = 13²
(8, 15, 17)	255 = 15×17	136 = 8×17	120 = 8×15	289 = 17²
(7, 24, 25)	600 = 24×25	175 = 7×25	168 = 7×24	625 = 25²
(20, 21, 29)	609 = 21×29	580 = 20×29	420 = 20×21	841 = 29²
All 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 theorem^[1] or the upside down Pythagorean theorem^[2]) is as follows:^[3]

Let

A

,

B

be the endpoints of the hypotenuse of a right triangle

△ ABC

. Let

D

be the foot of a perpendicular dropped from

C

, the vertex of the right angle, to the hypotenuse. Then

{\frac {1}{CD^{2}}}={\frac {1}{AC^{2}}}+{\frac {1}{BC^{2}}}.

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.

^ R. B. Nelsen, Proof Without Words: A Reciprocal Pythagorean Theorem, Mathematics Magazine, 82, December 2009, p. 370
^ The upside-down Pythagorean theorem, Jennifer Richinick, The Mathematical Gazette, Vol. 92, No. 524 (July 2008), pp. 313-316
^ Johan Wästlund, "Summing inverse squares by euclidean geometry", http://www.math.chalmers.se/~wastlund/Cosmic.pdf, pp. 4–5.

[1] R. B. Nelsen, Proof Without Words: A Reciprocal Pythagorean Theorem, Mathematics Magazine, 82, December 2009, p. 370

[2] The upside-down Pythagorean theorem, Jennifer Richinick, The Mathematical Gazette, Vol. 92, No. 524 (July 2008), pp. 313-316

[3] Johan Wästlund, "Summing inverse squares by euclidean geometry", http://www.math.chalmers.se/~wastlund/Cosmic.pdf, pp. 4–5.

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