{{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="max-width: 35em; text-align:right; float:right; clear:right; margin-left:1ex;" ! Base triple !! ''AC'' !! ''BC'' !! ''CD'' !! rowspan="6" style="padding:1px;"| !! ''AB'' |- | {{nobr|(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> |- | {{nobr|(5, 12, 13)}} || '''156''' = 12×13 || '''65''' = {{fsp}}5×13 || '''60''' = {{fsp}}5×12 || '''169''' = 13<sup>2</sup> |- | {{nobr|(8, 15, 17)}} || '''255''' = 15×17 || '''136''' = {{fsp}}8×17 || '''120''' = {{fsp}}8×15 || '''289''' = 17<sup>2</sup> |- | {{nobr|(7, 24, 25)}} || '''600''' = 24×25 || '''175''' = {{fsp}}7×25 || '''168''' = {{fsp}}7×24 || '''625''' = 25<sup>2</sup> |- | {{nobr|(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 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]], 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]] of a [[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 [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', the [[Pythagorean theorem#Converse|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>[[File:Inverted pythagorean theorem.png|thumb|320x320px|Visual proof]] as above.

Note in particular: :<math>\begin{align} \tfrac{1}{2} AC \cdot BC &= \tfrac{1}{2} AB \cdot CD \\[4pt] CD &= \tfrac{AC \cdot BC}{AB} \\[4pt] \end{align}</math>

==Special case of the cruciform curve== The [[cruciform curve]] or cross curve is a [[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]]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]] 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}}

{{Geometry-stub}} [[Category:Geometry]]