Inverse Pythagorean theorem

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] 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 △ABC can be expressed in terms of either AC and BC, or AB and CD: given CD > 0, AC > 0 and BC > 0.

Substituting x with AC and y with BC gives Inverse-Pythagorean triples can be generated using integer parameters t and u as follows.

[4] If two identical lamps are placed at A and B, the theorem and the inverse-square law imply that the light intensity at C is the same as when a single lamp is placed at D.

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Comparison of the inverse Pythagorean theorem with the Pythagorean theorem using the smallest positive integer inverse-Pythagorean triple in the table below.