Binary Star system common period

[slft]

Homework Statement

The questions are about binary star system.

Homework Equations

Why the two stars in a binary star system are always diametrically opposite positions?
Why the two stars have common period?
Why the inner star is more massive?

The Attempt at a Solution

For the period question, I'm thinking that they have same angular velocity, but I know that I should use force to explain why, but I don't know how to explain using gravitational force.
For the mass question, I think it's because of conservation of momentum, but don't know if it is right or how to explain.

 

[Fightfish]

Let us look at the definition of a binary star system: "a star system consisting of two stars orbiting around their common center of mass" according to Wikipedia. This succinct description in fact encodes all that you need to know to solve the question.
1. If the two stars are not diametrically opposite each other, what is the direction of the gravitational force acting on each of them? Can it possibly allow them to undergo circular motion?
2. This is linked to the first question. If they do not have the same period, then they would not always be diametrically opposite each other.
3. Based on the above two ideas, you can easily prove this using the fact that centripetal force acting on each is the gravitational force acting on it.