Problem of the Day #38

POTD January 12, 2023
For \triangle ABC with all of its points on the parabola y=x^2, let \delta( \triangle ABC) be the sum of the slopes of lines AB, BC, and CA.

Rosalind constructs a \triangle XYZ such the slope of \overline{XY} is twice the slope of \overline{YZ}, and the sum of the x-coordinates of points X and Z is \frac{2023}{\sqrt{2}}. Furthermore, over all possible configurations of \triangle XYZ, her construction is the one for which \tan(\angle XYZ) is maximized. Find the last three digits of \delta(\triangle XYZ)^2.