My question pertains to the question and solution below. In particular, can someone prove why $\tan \alpha_i = -t_i$ for each $i$? I think it would be worth clarifying what the angle each normal makes with the axis of the parabola (the x-axis) is. I think it's the smaller of the two angles formed by the normal lines and the axis.
If you have already been able to understand how the normal equation at $(at^2, 2at)$ is derived, then you understand that $y=(–t)x + at^3 + 2at$ has slope “-t”.
Also, by “angles which these three normals make with the axis”, the reference is not to the smaller of the angles, but only the angle between the normal and the positive direction of the x-axis. That is why tanα = slope, otherwise we would have tan($\pi — α$) = slope for lines making obtuse angles with +X, because slope is defined as the tangent of the angle between the normal and the positive direction of the x-axis, measured anti-clockwise with the +X axis as the reference line.