It is known and I have found proof of it (like here https://math.stackexchange.com/questions/635670/show-that-the-least-squares-line-must-pass-through-the-center-of-mass) that in Simple Linear Regression, the regression line always goes through the center of mass. But I have not find any proof that generalize this to the case where we have a generic number of covariables $p$. Does it hold? And if yes is there a proof?

It is known and I have found proof that (like here https://math.stackexchange.com/questions/635670/show-that-the-least-squares-line-must-pass-through-the-center-of-mass) in Simple Linear Regression, the regression line always goes through the center of mass. But I have not found any proof that generalizes this to the case where we have a generic number of covariables $p$.

Does it hold? And if yes is there a proof?