Hi. I recently made a thread having to do with this subject (link =
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). At the conclusion of my thread, I came away with the opinion that the sharringan and Byakugan were genetically similiar and that they probably had a common ancestor somewhere along the lines.
I think that each dojutsu (sharingan, rinnegan, and byakugan) have to be similiar in some way. After all, the definition of dojutsu is a technique using the eyes.
As for their relationships, the sharingan is directly related to the rinnegan. But the relationships with the Byakugan are a little more fuzzy. In my personal the uchiha and hyuga are decended from the same son of the sage. I think that similiar to how the senju and uzumaki are distantly related, the uchiha and hyuga are distantly related.
Those rumors came from the ignorant Kakashi who believes that the Sage Of The Six Paths is just a fairy tale. He's fodder. What would he know? There is no OP Hyuuga. There never will be. If Sharingan evolves five times, and the Byakugan doesn't, how can the Byakugan be better? Don't be silly.
Regarding your statement, I wanted to point out that Kakashi is not ignorant. He stated the Fact that there was a rumor that the sharingan was decended from the Uchiha. Also, there had not been any actual proof that the sage existed (at the point that Kakashi expressed his opinion about the sage).
We do now know that the sage existed now for a fact (confirmed by flashbacks by the tailed beasts - up until then we had been taking various people's word for it) but at that time, there was no proof (he believed that the rinnegan could have been a mutation).
To draw a parallel to the sage is to look at a religious figure such as Jesus (I AM NOT LOOKING TO GET INVOLVED IN A RELIGIOUS DEBATE). Christians believe that Jesus was the son of God and that he preformed various holy acts that were imposible for humans to preform (walking on water). People of other religions (or of no religion) deny either Jesus's existance or that he was the son of god. (I just stated the facts - I tried to remain completely objective)
Now if you look at the sage, and the actions that he accomplished, they do not seem resonable of a normal human (i.e. sealing of 10 tails into himself, splitting up the 10 tails into the 9 tailed beasts, creating the moon, etc.). It is very reasonable that some people denied his existance because they lacked proof.
I apoligize for going off topic.