$(ab+bc+ca)^3=abc(a+b+c)^3$, prove that $a,b,c$ are in $G.P.$

Question

Suppose $a,b,c$ are non-zero real numbers such that $$(ab+bc+ca)^3=abc.(a+b+c)^3$$ Prove that $a,b,c$ must be terms of a $G.P.$

I simplified this equation too $$(ab)^3+(bc)^3+(ca)^3=abc.(a^3+b^3+c^3)$$ I tried to subtract $3(abc)^2$ from both sides and it gave a factorised form. $$(ab+bc+ca).((ab)^2+(bc)^2+(ca)^2-abc.(a+b+c))=abc.(a+b+c).(a^2+b^2+c^2-ab-bc-ca)$$ How can I proceed after that?

Answer 1

This one is a long-way of solving (With a very obvious result). \begin{gather*} \implies(ab+bc+ca)^3-abc(a+b+c)^3=0\\ \end{gather*} \begin{multline*} \implies (a^3b^3+b^3c^3+c^3a^3+3a^2b^3c+3a^2bc^3+3a^3b^2c+3ab^2c^3+3a^3bc^2+3ab^3c^2)-\\abc(a^3+b^3+c^3+3a^2b+3ab^2+3b^2c+3bc^2+3c^2a+3ca^2)=0 \end{multline*}

After simplifying \begin{equation} \implies a^3b^3+b^3c^3+c^3a^3-a^4bc-ab^4c-abc^4=0 \end{equation} Factorising this will give \begin{equation} \implies(ab-c^2)(bc-a^2)(ca-b^2)=0 \end{equation} Therefore, a,b,c must be in G.P.

Answer 2

There was a good answer posted by @Albusdumbledore that was deleted. I am just reproducing that answer and will delete mine if the other is undeleted.

The suggestion was to write $a = q, b = q r$ and $c = q r k$. Naturally, since $a,b,c \ne 0$, we also have $q,r,k \ne 0$. Plugging this into the relation, one concludes that $r, k$ must satisfy the relation $$ (k-r) \left(k^2 r-1\right) \left(k r^2-1\right)=0 $$

All solutions lead to GP's. More specifically,

Case 1. If $k=r$, $a=q$, $b = r q$ and $c= r^2 q$ are consecutive terms in a GP with ratio $r$.

Case 2. If $k=\frac{1}{r^2}$ then $c=\frac{1}{r^2} q$, $a = q$ and $b=rq$ are (nonconsecutive) terms in a GP with ratio $r$

Case 3. If $k=\frac{1}{\sqrt{r}}$ then $c= \frac 1r q$, $a=q$ and $b= rq$ are (nonconsecutive) terms in a GP with ratio $\sqrt{r}$.

Case 4. If $k=-\frac{1}{\sqrt{r}}$ then $a=q$, $c=-\sqrt{r} q$, $b = rq$ are consecutive terms in a GP with ratio $-\sqrt{r}$