True or false the center of the circumscribed circle about a triangle is equidistant to the vertices of the inscribed triangle

True

Step-by-step explanation:

Consider triangle ABC inscribed in the circle with center at point O. Point O is a point of intersection of perpendicular bisectors of sides AC, AB and BC. According to the attached diagram, points E, F and G are midpoints of sides BC, AC and AB, respectively.

Then

Since segments OE, OF and OG are perpendicular bisectors of sides BC, AC and AB, then

You get three pairs of congruent triangles:

This gives you that

Each of these segments is a radius of the circumscribed circle about a triangle ABC, then the center of the circumscribed circle about a triangle is equidistant to the vertices of the inscribed triangle.