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 or the upside down Pythagorean theorem) is as follows:

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.