Deugging refraction in a sphere

Here is a setup to help debugging refraction using a sphere and a particular ray.  I have do track single rays like this every time I implement refraction.   Here is the setup:

So let's put a sphere of radius 1 the with center (0,0,0).  Now all of the action is in XY so I will leave off the Zs.   First, what is A?

If it is to hit where the surface of the sphere is at 45 degrees, it has to have y component cos(45) = sqrt(2)/2 .  So the origin of the ray should be o = (-something, sqrt(2)/2).  And v = (1, 0).  B, the reflected direction should have origin (-sqrt(2)/2, sqrt(2)/2) and direction (0,1).

What about C?  It has origin  (-sqrt(2)/2, sqrt(2)/2) and direction (1+sqrt(2)/2, -sqrt(2)/2) which has a length of about 1.84776.  So C = (0.9239, -0.3827) approximately.

But what refractive index produces such a C?   We an see that two of the sides have length 1, so the two angles are the same, and we can see that sin(theta) = (sqrt(2)/2) / 1.8776 = 0.38268.  So the refractive index must be the ratio of that ans sin(45).  So refractive_index = 1.8478.

So what is D?  By symmetry it must leave at 45 degrees so D = (1, -1) / sqrt(2).

And finally E reflects across the X axis and E  = (-0.9239, -0.3827) approximately