if an arc is given , then how we can prove that we can make the full circle
Take any three points on the arc say, A,B and C.Suppose B lies between A and C.
Join AB and BC
Construct the perpendicular bisector of the line segments AB and BC.
Since these three points are non collinear, their perpendicular bisectors have to meet.So suppose that they meet in M.
We know that every point on the perpendicular bisector of a line segment is equidistant from the end points of the line segment.
So , we can say that M is equidistant from A and B as well as B and C.
So M is a point which is equidistant from three non collinear points A,B and C.
So M must be the centre of the circle passing thru' hese three points.
Now having located the centre M, you can take either of MA,MB or MC as the radius and complete the circle.