Isosceles triangles from 2019

The length of the base side must be odd call it $B$ , so $2019-B$ is even and then the legs can be equal .

But $B$ can get values up to $1009$ , because the sum of each two sides in triangle must be greater than the last side .

So $B$ can be $1,3,...,1007,1009$ .. we have $\boxed{505}$ choices .

The sides of the triangles will be :

$(1,1009,1009)-(3,1008,1008)-...-(1007,506,506)-(1009,505,505)$