Circle-Circle intersection point article (it's VERY short)

I understood everything up until the very end :

x[sub]3[/sub] = x[sub]2[/sub] +- h ( y[sub]1[/sub] - y[sub]0[/sub] ) / d
y[sub]3[/sub] = y[sub]2[/sub] -+ h ( x[sub]1[/sub] - x[sub]0[/sub] ) / d

why the "y[sub]1[/sub] - y[sub]0[/sub]" and the negative h to get x[sub]3[/sub]?

(-(y1-y0), x1-x0)/d is a unit-length vector perpendicular to the line passing through the centers of the circle. P[sub]3[/sub] is the midpoint (sort of) of the overlap, and h is half the length of the cord bisecting (sort of) the overlap. Together, these values can be used to compute the intersection points.

Does that help at all?

(-(y1-y0), x1-x0)/d is a unit-length vector perpendicular to the line passing through the centers of the circle. P[sub]3[/sub] is the midpoint (sort of) of the overlap, and h is half the length of the cord bisecting (sort of) the overlap. Together, these values can be used to compute the intersection points.

Does that help at all?

Actually. Yes it does. I forgot that swapping the coordinates and negating one of them of a vector creates another vector perpendicular to it.

Also two more things, just out of curiosity if I took the circle formula :

(x - h[sub]1[/sub])[sup]2[/sup] + (y - c[sub]1[/sub])[sup]2[/sup] = r[sub]1[/sub][sup]2[/sup]
(x - h[sub]2[/sub])[sup]2[/sup] + (y - c2)[sup]2[/sup] = r[sub]2[/sub][sup]2[/sup]

and solve this system of equations by solving for x and then substituting it back into either of those and solve for y, would I get the same result? In other words, is that a valid way to find the intersection?

And doing this in 3D, I don't understand how he got
r^2 = R1^2 - (a^2+b^2+c^2)*t0^2for the radius of the intersecting circle between two spheres.

Thanks.