Given triangle ABC. On the bisector of angle A, find a point equidistant from vertices B and C.

Let the required point be point O.

Since, by condition, point O is equidistant from the vertices A and B of triangle ABC, then OA = OC, which means that triangle OAC must be isosceles.

Since in an isosceles triangle, the median drawn from the vertex of the angle opposite to the base is the height of the bisector, then we mark point H as the middle of the segment AC and draw a perpendicular to AC, until it intersects with the bisector BM.

The intersection point O is our desired point, and the triangle AOC is isosceles, OA = OC.