With a given centre and a given radius, only one circle can be drawn.
Idea: A circle is the set of all points that are at the same distance from one fixed point.
Step 1: Fix the centre.
( ext{Centre} = O )
Step 2: Fix the radius (the distance from the centre to the circle).
( ext{Radius} = r )
Step 3: A point (P) lies on the circle if its distance from (O) is exactly (r).
( OP = r )
Why only one circle?
Suppose there are two different circles with the same centre and the same radius.
( C_1 ) and ( C_2 ) with centre ( O ) and radius ( r )
Take any point (P) on (C_1). By definition of the circle:
( OP = r )
But every point whose distance from (O) is (r) must also lie on (C_2). So (P) is on (C_2) as well.
In the same way, any point on (C_2) is also on (C_1). Therefore, both circles have exactly the same points.
( C_1 = C_2 )
Conclusion: With a given centre and a given radius, the circle is uniquely determined. So only one circle can be drawn.