Three positive numbers a, b, c satisfy $a^2 + b^2 = c^2$; is it necessarily true that there exists a right triangle with side lengths a,b and c? If so, how could you go about constructing it? If not, why not?
I am new to proofs and I was reading a book where they posed this question. I understand that if we are given any right triangle, the relation between sides a, b, c is given by pythagorus' theorem; although, given $a^2 + b^2 = c^2$, does it necessarily imply that there exists a triangle with sides a, b, c?

Yes indeed the converse for Pythagoras is also true. When such a construction is made (SSS) the relation is satisfied. The vertices lie on a semicircle, and by Thale's thm a right angle is enclosed.
