It's not a theorem, and those 1/3 ratios are not set in stone. However ...
1) you want a reasonable signal swing at the output
2) you want a reasonable voltage on the emitter for stable biassing
(1) says make VCE = VRC
So to draw up a table of the options meeting (1), we have , with voltages given as a fraction of VCC
Like most compromises, anything near one end or the other is likely to be sub-optimal. So we go for somewhere in the middle. In the VE = 0.2 to 0.4 region, we have an output swing not far short of the maximum 0.5 possible, and we have plenty of VE, for stable biassing.
But most people like a number, rather than a range, as a range still leaves some choice. While providing a number, we pick a small number, to reflect that its exact value doesn't really matter. So we don't bias to 0.333VCC, we bias to 'about 1/3rd'.
Why choose 10% of Ic to go down the bias chain? Most small signal BJTs have a beta > 100, so the current error is going to be <10% of the total. This form of biassing with an emitter resistor is very tolerant of small errors. For almost all purposes, this is accurate enough for a final design, given that there will be unavoidable variations in temperature, VBE and beta amounting to this order of error anyway. If we want better control of Ic for some reason, then we monitor it and set the bias with an op-amp control loop.