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Question

# Prove that there is one and only one circle passing through three given points.

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Solution

## 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)OQ = OR (Since O lies on the perpendicular bisector of QR)∴ OP = OQ = ORLet OP = OQ = OR = r (Radius of a circle)Now, draw a circle C (O, r) with O as center and r as radius.Then, circle C(O, r) passes through the points P, Q and R.​We have to show that this circle is the only circle passing through the points P, Q and R.If possible, suppose there is another circle C(O′, t) which passes through the points P, Q and 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.∴ O ′ must coincide with the point O. (Since, two lines can not intersect at more than one point)As, O′P = t , OP = r and O ′ coincides with O,​t = r∴ C(O, r) and C(O', t) are congruent.Hence, there is one and only one circle passing through the given non-collinear points.

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