Relation between the side lengths and altitude of a right triangle
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
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 △ABC can be expressed in terms of either AC and BC, or AB and CD:
where the two parameters determining the shape of the curve, a and b are each CD.
Substituting x with AC and y with BC gives
Inverse-Pythagorean triples can be generated using integer parameters t and u as follows.[4]
Application
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.