The points of concurrency like circumference, orthocenter, incenter and centroid are important aspects of triangles. The perpendicular bisectors of the triangle form one of these points.
The perpendicular bisectors of the sides of the triangle are concurrent. The point of concurrency is called the circumcenter of the triangle.
In the adjoining diagram, the perpendicular bisectors of sides AB, BC and CA meet at point O.
Since O is a point on the perpendicular bisector of AB, it is equidistant from A and B. By similar arguments it is also equidistant from B and C as well. This makes the point O equidistant from all the three vertices of the triangle A,B and C.
Hence the circle drawn with O as center and OA as radius will pass through all the three vertices A, B and C. This circle is called the circumcircle of the triangle and hence the point O is known as the circumcenter of the triangle ABC.
In the case of an equilateral triangle all the four points of concurrency the circumcenter, orthocenter, incenter and centroid are coincident.Perpendicular bisectors in quadrilaterals
The diagonals of a rhombus or a square are perpendicular bisectors of each other. In the case of a kite the longer diagonal is the perpendicular bisector of the shorter.Perpendicular bisectors in circles
The center of a circle is the point concurrence of perpendicular bisectors of all chords of the circle. For a given chord the center of the circle lie on the perpendicular bisector of the chord, In this context the perpendicular bisector of a line segment can also be viewed as the locus of centers of all circles for which the line segment is a chord.
Two mutually perpendicular diameters of a circle bisect each other.