Inverse Pythagorean theorem

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

:Let , be the endpoints of the hypotenuse of a right triangle . Let be the foot of a perpendicular dropped from , the vertex of the right angle, to the hypotenuse. Then
::$\frac 1 = \frac 1 + \frac 1 .$

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 :
:\begin
\tfrac AC \cdot BC &= \tfrac AB \cdot CD \\ pt (AC \cdot BC)^2 &= (AB \cdot CD)^2 \\ pt \frac &= \frac
\end
given , and .

Using the Pythagorean theorem,
:\begin
\frac &= \frac \\ pt &= \frac + \frac \\ pt\quad \therefore \;\; \frac &= \frac + \frac
\end
as above.

Note in particular:
:\begin
\tfrac AC \cdot BC &= \tfrac AB \cdot CD \\ pt CD &= \tfrac \\ pt\end

Special case of the cruciform curve

The cruciform curve or cross curve is a quartic plane curve given by the equation
:$x^2 y^2 - b^2 x^2 - a^2 y^2 = 0$
where the two parameters determining the shape of the curve, and are each .

Substituting with and with gives
:\begin
AC^2 BC^2 - CD^2 AC^2 - CD^2 BC^2 &= 0 \\ ptAC^2 BC^2 &= CD^2 BC^2 + CD^2 AC^2 \\ pt\frac &= \frac + \frac \\ pt\therefore \;\; \frac &= \frac + \frac
\end

Inverse-Pythagorean triples can be generated using integer parameters and as follows.
:$\begin AC &= (t^2 + u^2)(t^2 - u^2) \\ BC &= 2tu(t^2 + u^2) \\ CD &= 2tu(t^2 - u^2) \end$

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 .

See also

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References

{{Geometry-stub
Geometry