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[parent] Garfield's proof of Pythagorean theorem (Proof)

James Garfield, the $ 20^{\mathrm{th}}$ president of the United States, gave the following proof of the Pythagorean Theorem in 1876. Consider the following trapezoid (note that this picture is half of the diagram used in Bhaskara's proof of the Pythagorean theorem).


\begin{pspicture}(-.5,-.5)(7.5,4.5) \rput(0,4){.} \psset{fillstyle=solid}% \pspo... ...,.25)(7,.25) \qline(3.2,.15)(3.05,.35) \qline(3.05,.35)(2.85,.2) \end{pspicture}

Recall that the area of a trapezoid with two parallel sides (in this case, the left and right sides) $ s_1$ and $ s_2$ and height $ h$ is

$\displaystyle h\frac{s_1+s_2}{2}$
So the area of the trapezoid above is
$\displaystyle (a+b)\frac{a+b}{2}=\frac{(a+b)^2}{2}$

The area of the yellow triangle (and that of the blue triangle) is

$\displaystyle \frac{ab}{2}$
while the area of the red triangle (also a right triangle) is
$\displaystyle \frac{c^2}{2}$

The two areas must be equal, so

$\displaystyle \frac{(a+b)^2}{2}$ $\displaystyle =2\frac{ab}{2}+\frac{c^2}{2}$    
$\displaystyle \frac{a^2+2ab+b^2}{2}$ $\displaystyle =ab + \frac{c^2}{2}$    
$\displaystyle a^2+2ab+b^2$ $\displaystyle =2ab+c^2$    
$\displaystyle a^2+b^2$ $\displaystyle =c^2$    



"Garfield's proof of Pythagorean theorem" is owned by rm50.
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Cross-references: right triangle, triangle, height, right, sides, parallel, area, trapezoid, Pythagorean theorem, United States
There are 2 references to this entry.

This is version 7 of Garfield's proof of Pythagorean theorem, born on 2007-05-25, modified 2007-05-28.
Object id is 9470, canonical name is GarfieldsProofOfPythagoreanTheorem.
Accessed 7340 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
Pending Errata and Addenda
None.
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Drawing right angles by rm50 on 2007-05-26 09:51:03
In response to pahio's correction, I've managed to draw right angles on the left and right hand triangles, but I can't get xypic to work for the third triangle. (I'm trying to learn xypic as I go). Perhaps someone can help. I'm trying
\[
\begin{xy}
,(0,24)
;(42,18)**@{-}
;(42,0)**@{-}
;(42,2)**@{-}
;(40,2)**@{-}
;(40,0)**@{-}
;(42,0)**@{-}
;(0,0)**@{-}
;(0,2)**@{-}
;(2,2)**@{-}
;(2,0)**@{-}
;(0,0)**@{-}
;(0,24)**@{-}
;(18,0)**@{-}
;(19.4,.94)**@{-}
;(18.46,2.34)**@{-}
;(17.06,1.4)**@{-}
;(18,0)**@{-}
;(42,18)**@{-}
,(9,-2)*{a}
,(44,9)*{a}
,(-2,12)*{b}
,(30,-2)*{b}
,(9,16)*{c}
,(30,12)*{c}
\end{xy}
\]

but nothing gets drawn for the elements using nonintegral coordinates. Is there a different way to do this?

Thanks,
Roger
[ reply | up ]
Half of Bhaskara's proof by rspuzio on 2007-05-26 00:18:57
It might be worth noting that the diagram used in this proof
is half of the diagram used in Bhaskara's proof of the same
result.
[ reply | up ]
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