Inverse Pythagorean theorem

In geometry, the inverse 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.

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.