We are asked to prove that given three positive numbers a, b and c, such that
$$ a < b + c\\ b < a + c\\ c < a + b, $$
then there exists a triangle whose sides have lengths $a,b$ and $c$.
They call this the Side-Side-Side Triangle Construction, and we just proved the Pythagorean Theorem, so we're advised to use it here.
No circumferences can be made at this point in the textbook.
Edit: I had written the inequalities wrong as someone pointed out. I meant that each one of the sides is less than the sum of the other two. Also, a, b and c, are positive real numbers.
Consider side a, draw a line equal to it's length and call the 2 endpoints B and C; then using your compass, draw a circle with origin C and radius b (call it C1) and another circle with origin B and radius c (call it C2); now I claim that there are 2 intersections between C1 and C2,because the sum of the 2 radius's (a and b) is greater than a (if we had b + c < a, then there wouldn't be any intersections). Call these 2 points of intersection P and Q, then any of the triangles PBC or QBC represent a triangle with sides a, b, c (due to the way we constructed them)