Prove that any three points on a circle cannot be collinear.


Let O be centre of the circle. Let P, Q, R be any points on the circle.
To prove: P,Q ,R cannot be collinear.
OP = OQ [Radii of same circle]

O is equidistant from end points P and Q of seg PQ.
O lies on perpendicular bisector of PQ. [Perpendicular bisector theorem]
In the same way we can prove that point O lies on perpendicular bisector of QR.
Point O is the point of intersection of perpendicular bisectors of PQ and QR….(i)

Imagine that the points P, Q, R are collinear.
Perpendicular bisector of PQ and QR will be parallel since perpendiculars to same line are parallel.
So the perpendicular bisector will not intersect at O.
This contradicts to statement (i) that perpendicular bisector intersect each other at O.
The imagination that P,Q,R are collinear is wrong.
Points P,Q,R are not collinear.

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