Theorem: There is one and only one circle passing through three given non-collinear points.

Given: Three non collinear points P, Q and R

To prove: There is one and only one circle passing through the points P, Q and R.

Construction: Join PQ and QR.

Draw perpendicular bisectors AB of PQ and CD of QR. Let the perpendicular bisectors intersect at the point O.

Now join OP, OQ and OR.

A circle is obtained passing through the points P, Q and R.

Proof: We know that, each and every point on the perpendicular bisector of a line segment is equidistant from its ends points.

Thus, OP = OQ [Since, O lies on the perpendicular bisector of PQ]

and OQ = OR. [Since, O lies on the perpendicular bisector of QR]

So, OP = OQ = OR.

Let OP = OQ = OR =r.

Now, draw a circle C(O,r) with O as centre andras radius.

Then, circle C(O,r) passes through the points P, Q and R.

Next, we prove this circle is the only circle passing through the points P, Q and R.

If possible, suppose there is a another circle C(O′,t) which passes through the points P, Q, R.

Then, O′ will lie on the perpendicular bisectors AB and CD.

But O was the intersection point of the perpendicular bisectors AB and CD.

So, O ′ must coincide with the point O.[Since, two lines can not intersect at more than one point]

As, O′P =tand OP =r; and O ′ coincides with O, we gett=r.

Therefore, C(O,r) and C(O,t) are congruent.

Thus, there is one and only one circle passing through three the given non-collinear points.

this was an alternative of this throrem.

please give me short answer

3 points are known to be collinear if they are in a straight line. But if we want a circle so two points will be collinear and can make a circle. But three collinear points cannot make a circle . Hence it is proved that any three points on a circle cannot be collinear.

Three points can't be collinear coz if they collide then there is a line and no circle is formed.