In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem or the upside down Pythagorean theorem) is as follows:
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.
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:
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.
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 .