Theorem 25.1- One and only one circle can pass through three non-collinear points

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Unit 25: Chords Of A Circle

Theorem 25.1:
One and only one circle can pass through three non-collinear points.

Given:
Three non-collinear points. say A, B and C.

To Prove:
One and only one circle can pass through A, B and C.

Construction:
Draw line segments AB and BC. Draw right bisectors ^<_ED^> and ^<_GF^> of AB and BC, respectively. ^<_ED^> and ^<_GF^> intersect at a point, say O. Draw OA,  OB and OC.

Proof:

Statemnets	Reasons
All points on <ED> are equidistant from A and B, so mOA = mOB ......... (i)	<ED> is the right bisector of AB, and O is a point on <ED>.
All points on <GF> are equidistant from B and C, so mOB = mOC ......... (ii)	<GF> is the right bisector of BC, and <GF> passess though O.
O is the unique point of intersection of <ED> and <GF> ....... (iii)	<ED> and <GF> are non-parallel lines.
The paint O is equidistant from A, B and C, i.e.  mOA = mOB = mOC = r, say ...... (iv)	From (i) and (ii)  Transitive property
The circle with centre only at O and radius r passes through A, B and C ...... (v)	OA, OB and OC are radial segment, and by (iii)
A, B and C are non collinear ..... (vi)	Given
Therefore, there exists one and only one From circle centered at O and with radius r passing through non-collinear points A, B and C.	From (iii), (iv), (v) and (vi)

Q.E.D

Text Book - Page No. 196 and 197.