Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 5 / a / Solution

Let a rational right triangle have legs $A,B$, hypotenuse $C$, and area $AB/2=D$. After interchanging the legs if necessary, setUsing $C^2=A^2+B^2$ and $AB=2D$ givesso $P=(x,y)$ lies on the congruent number elliptic curve $E_D:D{y}^2=x^3-x$. The triangle is nondegenerate, so $y\ne 0$ and $2P\ne O$.

Conversely the three displayed lengths in the question satisfyand their area isThus every such triangle is ${\Delta }_P$.