(i) Locus of point equidistant from three vertices :
Let in ΔABC, O is a moving point which is equidistant from three vertices A, B, and C.
∴ O is equidistant from A and B.
It means O is perpendicular bisector of AB
Again, O is equidistant from A and C.
It means O is perpendicular bisector of AC.
So O is perpendicular bisector of AC.
So, O is intersecting point of perpendicular bisectors of AB and AC.
Thus O is center of circle, passes through three vertices.
This circle passes through three vertices of triangle and we called it circumcenter of circle.
Hence required locus will be circumcenter of the circle.
(ii) Locus of a point equidistant from three sides :
Let O be equidistant from path P, Q, and R of sides BC, AC and AB respectively.
∴ O is equidistant from P and Q.
It means O is t bisector of PR.
Again O is equidistant from point R It means O is ⊥ bisector of PR.
So, O is intersecting point of PQ and PR.
Thus, O is centre of a circle touching three sides of triangle.
It means O is in centre.
Hence, required locus is in centre of circle.
