Let’s say we have a triple of integers $(a,b,c)$ which is assumed to be a Pythagorean triple, so that $$a^2+b^2=c^2.\tag{$\star$}$$ Without using ($\star$) directly or indirectly — but using other properties and theorems related to rational right triangles — what are ways of proving that $(a,b,c)$ is or is not a Pythagorean triple? I’m thinking of theorems on areas, medians, incircles and outcircles, triangle points, infinite ternary tree constructions, etc.
As a concrete example, consider the triple $(3,4,7)$. Without using the fact that $3^2+4^2 \ne 7^2$, how would one prove that $(3,4,7)$ is not a Pythagorean triple?
More to the point, consider the indeterminate triple $$(14x^3,(x^3-7)y^2, x^3y^2+7y^2-14x^3),\tag{$\dagger$}$$ which is derived from the Mordell equation $y^2=x^3+7$. That equation has no integer solutions (as proved using other algebraic methods) — I'm trying to find out if triangle properties can be used to prove the same result.