I read that you did not want the distance from B to A to equal the distance from A to C, but this diagrams shows exactly that. If you want to understand Spacetime diagrams, you should get familiar with the Minkowski metric, if you aren't already. This is the metric you have to use with spacetime diagrams.
It is just like a 4 dimension Euclidean metric ds2 = db2 + dx2 + dy2 + dz2, but you will see that the time vector is negative in this Minkowski metric which is ds2 = -dt2 + dx2 + dy2 + dz2. The negative sign only matters at angles more than 0 degrees from the point that you want to know the distance to because -dt2 = 0. Then in the case of spacetime diagrams, you just have ds2 = -dt2 + dx2. And in the case from B to A or from B to C it is just ds2 = 0 + dx2
So, as you see in your graph, from B to A and from B to C, you would use just the normal Euclidean/Pythagorean metric.
Knowing this will save you a lot of pain and frustration, especially as angles are more than 45 degrees. For angles more than 45 degrees, you have to totally through out all intuitive notions of distance and angles. For example, the angle that B and C make with F is not Tan(opposite/adjacent) like we were taught in school. And the distance from C to F is not found using Pythagoras a2 = b2 + c2, also like we were taught in school.