It is given , that AB=AC=AG. Now,if we draw a circle with centre A and AB as radius then,how should we prove that the circle will also pass through the points C and G?
What is the definition of circle in your geometry book?
It is going to be something like, "A circle is the set of all points a given distance from the center."
C,G are the given distance from the circle. They lie on circle A by definition.
The equation of the circle with center$(a, b)$ and radius $r$ is $(x-a)^2+(y-b)^2 = r^2$.
If this equation is satisfied by points A and B, then points A and C also satisfy this since the distances are the same; similarly for A and G.