# Triangle inequality question

LagrangeEuler
In the derivation of triangle inequality $$|(x,y)| \leq ||x|| ||y||$$ one use some ##z=x-ty## where ##t## is real number. And then from ##(z,z) \geq 0## one gets quadratic inequality
$$||x||^2+||y||^2t^2-2tRe(x,y) \geq 0$$
And from here they said that discriminant of quadratic equation
$$D=4(Re(x,y))^2-4 ||y||^2|x||^2 \leq 0$$
Could you explain me why ##<## sign in discriminant relation? When discriminant is less then zero solutions are complex conjugate numbers. But I still do not understand the discussed inequality. What about for example in case
$$||x||^2+||y||^2t^2-2tRe(x,y) \leq 0$$?

$$||x||^2+||y||^2t^2-2tRe(x,y) \geq 0$$