# Inverse Pythagorean theorem

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Relation between the side lengths and altitude of a right triangle

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

Base triple AC BC CD AB (3, 4, 5) 20 = 4× 5 15 = 3× 5 12 = 3× 4 25 = 52 (5, 12, 13) 156 = 12×13 65 = 5×13 60 = 5×12 169 = 132 (8, 15, 17) 255 = 15×17 136 = 8×17 120 = 8×15 289 = 172 (7, 24, 25) 600 = 24×25 175 = 7×25 168 = 7×24 625 = 252 (20, 21, 29) 609 = 21×29 580 = 20×29 420 = 20×21 841 = 292 All positive integer primitive inverse-Pythagorean triples having up to three digits, with the hypotenuse for comparison

In [geometry](/source/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](/source/Hypotenuse) of a [right triangle](/source/Right_triangle) △*ABC*. Let D be the foot of [a perpendicular](/source/Altitude_(triangle)) dropped from C, the vertex of the [right angle](/source/Right_angle), to the hypotenuse. Then - 1 C D 2 = 1 A C 2 + 1 B C 2 . {\displaystyle {\frac {1}{CD^{2}}}={\frac {1}{AC^{2}}}+{\frac {1}{BC^{2}}}.}

This theorem should not be confused with proposition 48 in book 1 of [Euclid](/source/Euclid)'s *[Elements](/source/Euclid's_Elements)*, the [converse](/source/Pythagorean_theorem#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:

- 1 2 A C ⋅ B C = 1 2 A B ⋅ C D ( A C ⋅ B C ) 2 = ( A B ⋅ C D ) 2 1 C D 2 = A B 2 A C 2 ⋅ B C 2 {\displaystyle {\begin{aligned}{\tfrac {1}{2}}AC\cdot BC&={\tfrac {1}{2}}AB\cdot CD\\[4pt](AC\cdot BC)^{2}&=(AB\cdot CD)^{2}\\[4pt]{\frac {1}{CD^{2}}}&={\frac {AB^{2}}{AC^{2}\cdot BC^{2}}}\end{aligned}}}

given *CD* > 0, *AC* > 0 and *BC* > 0.

Using the [Pythagorean theorem](/source/Pythagorean_theorem),

- 1 C D 2 = B C 2 + A C 2 A C 2 ⋅ B C 2 = B C 2 A C 2 ⋅ B C 2 + A C 2 A C 2 ⋅ B C 2 ∴ 1 C D 2 = 1 A C 2 + 1 B C 2 {\displaystyle {\begin{aligned}{\frac {1}{CD^{2}}}&={\frac {BC^{2}+AC^{2}}{AC^{2}\cdot BC^{2}}}\\[4pt]&={\frac {BC^{2}}{AC^{2}\cdot BC^{2}}}+{\frac {AC^{2}}{AC^{2}\cdot BC^{2}}}\\[4pt]\quad \therefore \;\;{\frac {1}{CD^{2}}}&={\frac {1}{AC^{2}}}+{\frac {1}{BC^{2}}}\end{aligned}}} Visual proof

as above.

Note in particular:

- 1 2 A C ⋅ B C = 1 2 A B ⋅ C D C D = A C ⋅ B C A B {\displaystyle {\begin{aligned}{\tfrac {1}{2}}AC\cdot BC&={\tfrac {1}{2}}AB\cdot CD\\[4pt]CD&={\tfrac {AC\cdot BC}{AB}}\\[4pt]\end{aligned}}}

## Special case of the cruciform curve

The [cruciform curve](/source/Cruciform_curve) or cross curve is a [quartic plane curve](/source/Quartic_plane_curve) given by the equation

- x 2 y 2 − b 2 x 2 − a 2 y 2 = 0 {\displaystyle x^{2}y^{2}-b^{2}x^{2}-a^{2}y^{2}=0}

where the two [parameters](/source/Parameter) determining the shape of the curve, a and b are each CD.

Substituting x with AC and y with BC gives

- A C 2 B C 2 − C D 2 A C 2 − C D 2 B C 2 = 0 A C 2 B C 2 = C D 2 B C 2 + C D 2 A C 2 1 C D 2 = B C 2 A C 2 ⋅ B C 2 + A C 2 A C 2 ⋅ B C 2 ∴ 1 C D 2 = 1 A C 2 + 1 B C 2 {\displaystyle {\begin{aligned}AC^{2}BC^{2}-CD^{2}AC^{2}-CD^{2}BC^{2}&=0\\[4pt]AC^{2}BC^{2}&=CD^{2}BC^{2}+CD^{2}AC^{2}\\[4pt]{\frac {1}{CD^{2}}}&={\frac {BC^{2}}{AC^{2}\cdot BC^{2}}}+{\frac {AC^{2}}{AC^{2}\cdot BC^{2}}}\\[4pt]\therefore \;\;{\frac {1}{CD^{2}}}&={\frac {1}{AC^{2}}}+{\frac {1}{BC^{2}}}\end{aligned}}}

Inverse-Pythagorean triples can be generated using [integer](/source/Integer) parameters t and u as follows.[4]

- A C = ( t 2 + u 2 ) ( t 2 − u 2 ) B C = 2 t u ( t 2 + u 2 ) C D = 2 t u ( t 2 − u 2 ) {\displaystyle {\begin{aligned}AC&=(t^{2}+u^{2})(t^{2}-u^{2})\\BC&=2tu(t^{2}+u^{2})\\CD&=2tu(t^{2}-u^{2})\end{aligned}}}

## Application

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

## See also

- [Geometric mean theorem](/source/Geometric_mean_theorem) – Theorem about right triangles

- [Pythagorean theorem](/source/Pythagorean_theorem) – Relation between sides of a right triangle

## References

1. **[^](#cite_ref-1)** R. B. Nelsen, Proof Without Words: A Reciprocal Pythagorean Theorem, Mathematics Magazine, 82, December 2009, p. 370

1. **[^](#cite_ref-2)** The upside-down Pythagorean theorem, Jennifer Richinick, The Mathematical Gazette, Vol. 92, No. 524 (July 2008), pp. 313-316

1. **[^](#cite_ref-3)** Johan Wästlund, "Summing inverse squares by euclidean geometry", [http://www.math.chalmers.se/~wastlund/Cosmic.pdf](http://www.math.chalmers.se/~wastlund/Cosmic.pdf), pp. 4–5.

1. **[^](#cite_ref-4)** ["Diophantine equation of three variables"](http://math.stackexchange.com/a/2688836).

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