The language that I am allowed to use: $$\mathcal L = (\{\{ +, \cdot\}, \{ = \}, \{ 0, 1 \} \}, \mathbb R) $$
This is my solution (I used the triangle inequality to express that $a+b > c \land a+c > b \land b+c > a$)
$$\alpha, \beta, \gamma - \mbox{Auxilary variables used to express that one quantity is greater than the other}$$
$$(\exists \alpha)(\exists \beta)(\exists \gamma)\left(a+ b = c + \alpha \alpha \land a+c = b + \beta \beta \land b + c = \gamma \gamma\right) \land \neg(a=0 \lor b =0 \lor c=0)$$
What do you think of my solution?
It's alright, although if you don't write $\alpha^2,\beta^2,\gamma^2$, you should also not use juxtaposition, and write $\alpha\cdot\alpha$ etc. To make it more clear, I would also ask that $\alpha,\beta$ and $\gamma$ are nonzero. I believe it follows, but it's not immediately obvious. A clever way of writing that is $\neg (\alpha\cdot\beta\cdot\gamma\cdot a\cdot b \cdot c=0)$
On a tangential note, your description of language looks strange. What you probably meant to write is that you are working in the structure $(\mathbf R,+,\cdot,=,0,1)$.