Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters

What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial $ax^2+bx+c$ has real roots if the parameters $a,b,c$ are exponentially distributed with common parameter. (The parameter was specified as $1$ but doesn't matter.)

This didn't seem to me like a natural question to ask, since the exponential distribution has nice properties with respect to addition and not so much with respect to multiplication, and the discriminant condition $b^2\ge4ac$ is multiplicative; so I was a bit surprised by the simple answer $\frac13$. This seems to cry out for a symmetric proof, e. g. using a transformation to three equivalent variables one of which must equiprobably be the greatest to fulfill the condition. However, I couldn't come up with such a transformation (I tried $\sqrt{ab}$, $\sqrt{bc}$, $\sqrt{ca}$ and $a-b$, $b-c$, $c-a$) and could only obtain the probability by performing an asymmetric and somewhat involved triple integration.

Can you provide a more elegant proof of this result?