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:
The triangle is equilateral
$MB$ is an angle bisector to the angle $B$
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?