$\Delta$ Allergy

Since $\alpha + \beta,\alpha^2 + \beta^2,\alpha^3 + \beta^3$ are in GP,

We have $(\alpha^2 + \beta^2)^2 = (\alpha + \beta)(\alpha^3 + \beta^3)$

$\alpha^4 + \beta^4 + 2\alpha^2\beta^2 = \alpha^4 + \beta^4 + \alpha\beta^3 + \alpha^3\beta$

$2\alpha^2\beta^2 = \alpha\beta^3 + \alpha^3\beta$

$\alpha\beta^3 + \alpha^3\beta - 2\alpha^2\beta^2 = 0$

$\alpha\beta^3 - \alpha^2\beta^2 - \alpha^2\beta^2 + \alpha^3\beta = 0$

$\alpha\beta^2(\beta - \alpha) - \alpha^2\beta(\beta - \alpha) = 0$

$(\beta - \alpha)( \alpha\beta^2 - \alpha^2\beta) = 0$

$\alpha\beta(\beta - \alpha)^2 = 0$

$\Rightarrow \alpha = 0, \beta = 0 , \text{or } \alpha = \beta$

Now, $\alpha,\beta = 0$ implies $c = 0,$ since $c = a\alpha\beta$ .

And $\alpha = \beta$ implies $\Delta = 0$ .

Combining them implies that $\boxed{c\Delta = 0}$