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geometric proof of Pythagorean triplet
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(Proof)
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If $x^2+y^2=z^2$ for $x,y,z$ positive integers is a pythagorean triple, then dividing through by $z^2$ , we can write this in the form $r^2+s^2=1$ for positive rational numbers $r,s$ . There is thus a 1-1 correspondence between primitive pythagorean triples (i.e.
those for which $x,y$ , and $z$ are pairwise coprime) and rational points in the first quadrant on the unit circle.
To find all such points on the unit circle, consider the following diagram:
The line from $A$ to $B$ is the line $y=t(x+1)$ ; we parametrize this as $t$ ranges over $[0,1]$ to capture all points in the first quadrant.
Substituting $y=t(x+1)$ back into the equation for the unit circle, we get
Solving for $x$ using the quadratic formula (or, alternatively, dividing this polynomial by the known factor $x+1)$ , and computing $y$ using the equation of the line, we get $$ x = \frac{1-t^2}{1+t^2}=1-\frac{2t^2}{1+t^2},\quad y = \frac{2t}{1+t^2} $$ So if both $x$ and $y$ are to be rational, we must have that both $$ \frac{2t^2}{1+t^2}\quad\text{and}\quad\frac{2t}{1+t^2} $$ are rational, and thus their quotient $t$ must be rational. Writing $t=\frac{q}{p}$ , we get $$ x = \frac{1-t^2}{1+t^2} = \frac{1-\left(\frac{q}{p}\right)^2}{1+\left(\frac{q}{p}\right)^2} = \frac{p^2-q^2}{p^2+q^2}, \quad y = \frac{2\frac{q}{p}}{1+\left(\frac{q}{p}\right)^2}=\frac{2pq}{p^2+q^2} $$ and then $x^2+y^2=1$ becomes $$ (p^2+q^2)^2 = (p^2-q^2)^2 + (2pq)^2 $$ which is the desired parametrization of the pythagorean triple.
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"geometric proof of Pythagorean triplet" is owned by rm50.
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Cross-references: quotient, factor, polynomial, quadratic formula, equation, ranges, line, diagram, unit circle, quadrant, points, rational, pairwise coprime, primitive Pythagorean triples, 1-1 correspondence, rational numbers, Pythagorean triple, integers, positive
This is version 3 of geometric proof of Pythagorean triplet, born on 2008-12-12, modified 2008-12-12.
Object id is 11337, canonical name is GeometricProofOfPythagoreanTriplet.
Accessed 392 times total.
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Pending Errata and Addenda
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