Legendre's Theorem for Non-Pairwise Coprime Coefficients?

Question

I want to solve the problem of existence of rational points on conic sections by using the theorem of Legendre (1785):

Fix squarefree and pairwise-coprime integers $a,b,c\in\mathbb{Z}$, not all of the same sign. Then there exist rational numbers $x,y\in\mathbb{Q}$ satisfying $ax^2+by^2+c=0$ if and only if ($-ba$ is square mod $c$), ($-ac$ is square mod $b$) and ($-bc$ is square mod $a$).

So far I have been able to reduce the problem to the case of $ax^2+by^2+c=0$ where $a,b,c\in\mathbb{Z}$ are squarefree integers, not all of the same sign, such that $\mathrm{gcd}(a,b,c)=1$.

Question: Can the case $\mathrm{gcd}(a,b,c)=1$ be reduced to the Legendre case $$\mathrm{gcd}(a,b)=\mathrm{gcd}(a,c)=\mathrm{gcd}(b,c)=1$$ or is there some idea missing?

Answer 1

Suppose two of $a$, $b$ and $c$ are divisible by some $d$, say $b=b'd$, $c=c'd$. Then $ax^2+by^2+c=ax^2+b'dy^2+c'd=0$ is soluble iff $(a/d)x^2+b'y^2+c'=0$ is soluble iff $adx^2+b'y^2+c'=0$ is soluble. Note $d>1$ means that $|(ad)b'c'|<|abc|$.