Each property, proved a step at a time. Every one of them starts by joining a point to the centre, because OA = OP = OB is the only fact doing any work.
Every proof here rests on OA = OP = OB.
Three radii, so the triangles are isosceles and their base angles match. After that it is only the exterior angle of a triangle, or the angle sum of one. Nothing else is doing any work.