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Question
Prove that any three points on a circle cannot be collinear.
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Solution
Let A, B and C be any three points on a circle. Suppose these three points A, B and C on the circle are collinear.
Therefore, the perpendicular bisectors of the chords AB and BC must be parallel because two or more lines which are perpendicular to a given line are parallel to each other.
Now, AB and BC are the chords of the circle. We know that the perpendicular bisector of the chord of a circle passing through its centre.
So, the perpendicular bisectors of the chords AB and BC must intersect at the centre of the circle.
This is a contradiction to our statement that the perpendicular bisectors of AB and BC must be parallel, as parallel lines do not intersect at a point.
Hence, our assumption that three points A, B and C on the circle are collinear is not correct.
Thus, any three points on a circle cannot be collinear.
Let A, B and C be any three points on a circle. Suppose these three points A, B and C on the circle are collinear.
Therefore, the perpendicular bisectors of the chords AB and BC must be parallel because two or more lines which are perpendicular to a given line are parallel to each other.
Now, AB and BC are the chords of the circle. We know that the perpendicular bisector of the chord of a circle passing through its centre.
So, the perpendicular bisectors of the chords AB and BC must intersect at the centre of the circle.
This is a contradiction to our statement that the perpendicular bisectors of AB and BC must be parallel, as parallel lines do not intersect at a point.
Hence, our assumption that three points A, B and C on the circle are collinear is not correct.
Thus, any three points on a circle cannot be collinear.
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