Question

What is the locus of points which are equidistant from three given non-collinear points $A,B$ and $C$? Justify your answer.

Answer 1

Let $A,B$ and $C$ be three non-collinear points

Join $AB$ and $BC$

Draw perpendicular bisector of $AB$ and $BC$

Let them meet at point $O$

Then $O$ lie on the right bisector of $AB$

So, $OA=OB$

And O lies on the right bisector of BC

So, $OB=OC$

Hence, the point $O$ is equidistant from $A,B$ and $C$.

Now since the right bisector of $AB$ and $BC$ are non-parallel lines, therefore they have only one point in common.

So, $O$ is the only point equidistant from $A,B$ and $C$.

Hence, the required locus is the centre of the circle through the given non-collinear points.