= Infinitely many rational right triangles from a nontorsion point
For $D>0$, a nontorsion <rational point> on the <congruent number elliptic curve> $y^2=x^3-D^2x$ gives infinitely many positive rational <right triangles> of area $D$. Apply the formulas $a=|(x^2-D^2)/y|$, $b=|2Dx/y|$, $c=|(x^2+D^2)/y|$ to its multiples. The map is finite-to-one: if $r=c/b$ is fixed, then $|x|$ satisfies $X^2-2DrX+D^2=0$, leaving only finitely many coordinate signs. Thus infinitely many points give infinitely many distinct triangles.
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