Why is this inscribed isosceles triangle equilateral?

The isosceles triangle $$ABC$$ is $$AC=AB$$. Its inscribed circle has radius one and center $$M$$ inside of the triangle. Show that the following statements are equivalent:

1. The triangle is equilateral
2. $$MB$$ is an angle bisector to the angle $$B$$
3. The angle $$BMC$$ is $$120^O$$

If 1. is true then I can see how the rest follows, but I can't get my mind to accept that the isosceles triangle "becomes" an equilateral as a general case. Could you help nudge my head towards that direction?

You are not being asked to prove (1).

You are only being asked to prove that if (1) is true then the others are true and vice-versa.

To reduce your work you can set your work out as three proofs

(1) implies (2)

(2) implies (3)

(3) implies (1).

Then all the statements are true if any one statement is true.