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

\frac 1  = \frac 1  + \frac 1 .

This theorem should not be confused with proposition 48 in book 1 of

Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...

'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'': :

\begin
   \tfrac AC \cdot BC    &= \tfrac AB \cdot CD
\\             (AC \cdot BC)^2 &=             (AB \cdot CD)^2
\\  \frac             &= \frac
\end

given ''CD'' > 0, ''AC'' > 0 and ''BC'' > 0. Using the Pythagorean theorem, :

\begin
                         \frac &= \frac
\\                                      &= \frac + \frac
\\ \quad \therefore \;\; \frac &= \frac + \frac
\end

as above.

Special case of the cruciform curve

The

cruciform curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree of a polynomial, degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0 ...

or cross curve is a

quartic plane curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one of ...

given by the equation :

x^2 y^2 - b^2 x^2 - a^2 y^2 = 0

where the two

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

s determining the shape of the curve, ''a'' and ''b'' are each ''CD''. Substituting ''x'' with ''AC'' and ''y'' with ''BC'' gives :

\begin
   AC^2 BC^2 - CD^2 AC^2 - CD^2 BC^2 &= 0
\\ AC^2 BC^2                         &= CD^2 BC^2 + CD^2 AC^2
\\ \frac                    &= \frac + \frac
\\ \therefore \;\; \frac    &= \frac + \frac
\end

Inverse-Pythagorean triples can be generated using integer parameters ''t'' and ''u'' 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 A and B, the theorem and the

inverse-square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understo ...

imply that the amount of light received at C is the same as when a single lamp is placed at D.

References

{{geometry-stub Geometry

Proof

Special case of the cruciform curve

Application

See also

References