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The map $g: B \to A, \ (x,y) \mapsto \left(\dfrac {x^2 – 25} y, \dfrac {10x} y, \dfrac {x^2 + 25} y \right)$ is a bijection where $A = \{ (a,b,c) \in \Bbb Q ^3 : a^2 + b^2 = c^2, \ ab = 10 \}$ and $B = \{ (x,y) \in \Bbb Q ^2 : y^2 = x^3 – 25x, \ y \ne 0 \}$.

Given the *Pythagorean triple*,

$$

\begin{aligned}

a&=\frac {x^2 – 25} y\\

b&=\frac {10x} y\\

c&=\frac {x^2 + 25} y

\end{aligned}

$$

so $a^2+b^2 = c^2$ and $y\neq 0$. We wish to set its *area* $G = 5$. Thus,

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$$G = \frac{1}{2}ab = \frac{1}2 \frac {10x} y \frac {(x^2 – 25)} y = 5$$

Or simply,

$$x^3-25x = y^2\tag1$$

Can you help find two non-congruent right-angled triangles that have rational side lengths and area equal to $5$?

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*Update*: It turns out that for rational $a,b,c$, if,

$$a^2+b^2 = c^2$$

$$\tfrac{1}{2}ab = n$$

then $n$ is a *congruent number*. The first few are $n = 5, 6, 7, 13, 14, 15, 20, 21, 22,\dots$

After all the terminology, I guess all the OP wanted was to solve $(1)$. He asks, “… *find two non-congruent right-angled triangles that have rational side lengths and area equal to 5*“.

We’ll define congruence as *“… two figures are congruent if they have the same shape and size.*” Since we want

$$a,b,c = \frac{3}{2},\;\frac{20}{3},\;\frac{41}{6}$$

$$a,b,c = \frac{1519}{492},\;\frac{4920}{1519},\;\frac{3344161}{747348}$$

However, since the elliptic curve,

$$x^3-25x = y^2\tag1$$

has an *infinite* number of rational points, then there is no need to stop at just two. Two small solutions to $(1)$ are $x_1 = 5^2/2^2 =u^2/v^2$ and $x_2 = 45$. We can use the first one as an easy way to generate an infinite subset of solutions. Let, $x =u^2/v^2$, so,

$$\frac{u^2}{v^6}(u^4-25v^4) = y^2$$

or just,

$$u^4-25v^4 = w^2\tag2$$

a curve also discussed in the post cited by J. Lahtonen. Here is a theorem by Lagrange. Given an initial solution $u^4+bv^4 = w^2$, then further ones are,

$$X^4+bY^4 = Z^2$$

where,

$$X,Y,Z = u^4-bv^4,\; 2uvw,\; (u^4+bv^4)^2 + 4 b u^4 v^4$$

Thus, using Lagrange’s theorem recursively, we have the infinite sequence,

$$u,v,w = 5,\;2,\;15$$

$$u,v,w = 41,\;12,\;1519$$

$$u,v,w = 3344161,\; 1494696,\; 535583225279$$

and so on.

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