Theorem 1. There is one and only one circle passing through three non-collinear points.
Given:- A, B and C are three non-collinear points
To prove:- One and only one circle are drawn through the points A, B and C.
Construction:- AB and BC are joined PL and QM are drawn perpendicular bisectors respectively of AB and BC to intersect at O.
OA, OB and OC are joined.
Proof:- O lie on PL, the perpendicular bisector of AB
OA = OB ........................(i)
similarty OB = OC...........................(ii)
from (i) and (ii)
OA = OB = OC = r, say
A circle is drawn taking O as centre and r as radius to pass through A, B and C.
Let there be another centre O’ and another radius s, of circle through A, B and C. O’ must lie on PL and QM. As two lines can intersect at only one point. Hence O and O’ coincide.
OA = O' A = r = s
Hence there is unique circle passing through A, B and C.
Other Educational Portals
English | cbse | Science | Mathematics
Comments