What's the right line? Not like fractions on the OB (viewed as a disc)...?
pj
chgo
The equal distances need to be on the GB-OB line. The three points are
Center of OB
Contact point
Center of GB
The distance between the first two points appears to be smaller than the distance between the second two because the points are farther from you. The error is not large for normal-length shots.
Draw out an overhead diagram with a half-ball shot and the cue ball half a ball from the object ball. The angle between the center of the cue ball and the contact point is less than half of the angle between the contact point and the edge of object ball ( through the center of the ghost ball). Most would consider that to not be "double the distance".
Often you can find imperfections in a system by taking the angle/positions to an extreme. If the system breaks there, you need to figure out under which conditions the system is "good enough" to use. And if the system breaks at an extreme it probably has errors for all cases.
Another case of a system that breaks is fractional ball if you only look at the first four values:
full ball -- 0 degrees cut
3/4 ball -- 15 degrees cut
1/2 ball -- 30 degrees cut
1/4 ball -- 45 degrees cut
A nice progression that clearly fails for the next case in that obvious sequence:
0 ball -- 60 degree cut -- off by 30 degrees
Of course it is easy to show that 3/4 and 1/4 are not exactly the angles shown above but without knowing trigonometry/geometry the fifth step shows you that you have to be careful and something is not quite right. (What is not right mathematically is that the sine of an angle is not equal to the angle but it is nearly equal for small angles.)