View source for Inverse Pythagorean theorem

{{Short description|Relation between the side lengths and altitude of a right triangle}}
[[File:inverse_pythagorean_theorem.svg|thumb|Comparison of the inverse Pythagorean theorem with the Pythagorean theorem {{nowrap|using the smallest}} positive integer inverse-Pythagorean triple in the table below. ]]
{| class="wikitable" style="text-align:right; float:right; clear:right; margin-left:1ex;"
! Base<br />Pytha-<br />gorean<br />triple !! ''AC'' !! ''BC'' !! ''CD'' !! rowspan="6" style="padding:1px;"| !! ''AB''
|-
|  (3, {{fsp}}4, {{fsp}}5) || '''20''' = {{fsp}}4×{{fsp}}5  ||  '''15''' = {{fsp}}3×{{fsp}}5 ||  '''12''' = {{fsp}}3×{{fsp}}4 ||  '''25''' = {{fsp}}5<sup>2</sup>
|-
|  (5,       12,       13) || '''156''' =       12×13       ||  '''65''' = {{fsp}}5×13       ||  '''60''' = {{fsp}}5×12       || '''169''' =       13<sup>2</sup>
|-
|  (8,       15,       17) || '''255''' =       15×17       || '''136''' = {{fsp}}8×17       || '''120''' = {{fsp}}8×15       || '''289''' =       17<sup>2</sup>
|-
|  (7,       24,       25) || '''600''' =       24×25       || '''175''' = {{fsp}}7×25       || '''168''' = {{fsp}}7×24       || '''625''' =       25<sup>2</sup>
|-
| (20,       21,       29) || '''609''' =       21×29       || '''580''' =       20×29       || '''420''' =       20×21       || '''841''' =       29<sup>2</sup>
|-
| colspan="6" style="text-align:left;"|All positive integer primitive inverse-Pythagorean triples having up to<br />three digits, with the hypotenuse for comparison<!--import fractions
def roundm(x): return int(round(x))
is_seens = set()
for  a in range(1, 999 + 1):
 for b in range(1, a   + 1):
  r  = fractions.Fraction(a, b)
  c2 = fractions.Fraction((a * a) * (b * b), ((a * a) + (b * b)))
  c  = roundm(c2.numerator ** 0.5)
  if c2.denominator == 1 and c ** 2 == c2.numerator and r not in is_seens:
   h = roundm((a * a + b * b) ** 0.5)
   print('| %4s   || %4s   || %4s   || %4s = %2s<sup>2</sup>\n|-' % (a, b, c, h, roundm(h ** 0.5)))
   is_seens.add(r)
-->
|}

In [[Geometry|geometry]], the '''inverse Pythagorean theorem''' (also known as the '''reciprocal Pythagorean theorem'''<ref>R. B. Nelsen, Proof Without Words: A Reciprocal Pythagorean Theorem, Mathematics Magazine, 82, December 2009, p. 370</ref> or the '''upside down Pythagorean theorem'''<ref>The upside-down Pythagorean theorem, Jennifer Richinick, The Mathematical Gazette, Vol. 92, No. 524 (July 2008), pp. 313-316</ref>) is as follows:<ref>Johan Wästlund, "Summing inverse squares by euclidean
geometry", http://www.math.chalmers.se/~wastlund/Cosmic.pdf, pp.&nbsp;4–5.</ref>

:Let {{mvar|A}}, {{mvar|B}} be the endpoints of the [[Hypotenuse|hypotenuse]] of a [[Right triangle|right triangle]] {{math|△''ABC''}}. Let {{mvar|D}} be the foot of [[Altitude (triangle)|a perpendicular]] dropped from {{mvar|C}}, the vertex of the right angle, to the hypotenuse. Then
::<math> \frac 1 {CD^2} = \frac 1 {AC^2} + \frac 1 {BC^2}.</math>

This theorem should not be confused with proposition 48 in book 1 of [[Biography:Euclid|Euclid]]'s ''[[Euclid's Elements|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 {{math|△''ABC''}} can be expressed in terms of either {{mvar|AC}} and {{mvar|BC}}, or {{mvar|AB}} and {{mvar|CD}}:
:<math>\begin{align}
\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{align}</math>
given {{math|''CD'' &gt; 0}}, {{math|''AC'' &gt; 0}} and {{math|''BC'' &gt; 0}}.

Using the [[Pythagorean theorem]],
:<math>\begin{align}
\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{align}</math>
as above.

==Special case of the cruciform curve==
The [[Cruciform curve|cruciform curve]] or cross curve is a [[Quartic plane curve|quartic plane curve]] given by the equation
:<math>x^2 y^2 - b^2 x^2 - a^2 y^2 = 0</math>
where the two [[Parameter|parameter]]s determining the shape of the curve, {{mvar|a}} and {{mvar|b}} are each {{mvar|CD}}.

Substituting {{mvar|x}} with {{mvar|AC}} and {{mvar|y}} with {{mvar|BC}} gives
:<math>\begin{align}
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{align}</math>

Inverse-Pythagorean triples can be generated using integer parameters {{mvar|t}} and {{mvar|u}} as follows.<ref>{{Cite web|url=http://math.stackexchange.com/a/2688836|title=Diophantine equation of three variables}}</ref>
:<math>\begin{align}
AC &= (t^2 + u^2)(t^2 - u^2) \\
BC &= 2tu(t^2 + u^2) \\
CD &= 2tu(t^2 - u^2)
\end{align}</math>
<!--
for  t in range(1, 5 + 1):
 for u in range(1, t):
  print(t, u, 2 * t * u * (t ** 2 + u ** 2),
        (t ** 2 - u ** 2) * (t ** 2 + u ** 2), 2 * t * u * (t ** 2 - u ** 2))
-->

==Application==

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

==See also==
* {{Annotated link|Geometric mean theorem}}
* {{Annotated link|Pythagorean theorem}}

==References==
{{Reflist}}


[[Category:Geometry]]

{{Sourceattribution|Inverse Pythagorean theorem}}