Circumcenter - Point of Concurrency of Perpendicular Bisectors

Consider any triangle with its vertices represented by three points in a coordinate system. Question is, can we find a point that is at the same distance from the vertices of the triangle?

And if such a point exists then is it unique for that triangle or are there more such points?

The answer to the first question is Yes. For any triangle, there exists a point in the plane of that triangle - inside or outside or lying on its edge - same distance away from the three vertices. This point is called the CIRCUMCENTER. Moreover, this point is unique for a given triangle, that is, a triangle has one and only one circumcenter.

In △MNP, Point C is the circumcenter &

CM = CP = CN

For acute angled triangles, the circumcenter is always present INSIDE of the triangle, and conversely, if circumcenter lies inside of a triangle, the triangle is acute. 

For obtuse angled triangles, circumcenter is always present OUTSIDE of the triangle, and likewise, if the circumcenter is outside of a triangle, the triangle is obtuse. 

In a right angled triangle the circumcenter is present ON the hypotenuse of the triangle. In fact, it is the midpoint of the hypotenuse in all right angled triangles. 

Why the name ‘Circumcenter’? 

Because it is center of the circle that circumscribes the triangle(circle through the three vertices). The circumscribing circle is called the CIRCUMCIRCLE of that triangle and its center is circum-center of that triangle. Circumcenter is unique for a given triangle and so is circumcircle.

Circumcenter is also known as the point where three perpendicular bisectors of a triangle intersect. A perpendicular bisector is a line that is perpendicular to a side of a triangle, and also divides the side into two halves of equal lengths. In any triangle, three perpendicular bisectors of three sides always intersect at a single point -  the circumcenter.