the point equidistant from all three points P, Q, and R is the intersection of the perpendicular bisectors of PQ, QR and PR

PQ is a vertical line ... the equation of the perpendicular bisector is y = -1/2

PR is a horzontal line ... the equation of the perpendicular bisector is x = 1/2

the point of intersection is (1/2, -1/2), which is equidistant from the three points.

fyi ... the point equidistant from the three vertices of a right triangle is always the midpoint of the hypotenuse ... make a sketch on graph paper and check it out.