 Circumcenter - Concurrent Bisectors MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts  A point of concurrency is the point where three or more lines intersect.

We will continue our investigation of the classical triangle centers with the circumcenter.

CIRCUMCENTER - concurrent perpendicular bisectors

The perpendicular bisectors of the sides of a triangle are concurrent (they intersect in one common point). The point of concurrency of the perpendicular bisectors of the sides is called the circumcenter of the triangle. The point of concurrency is not necessarily inside the triangle. It may actually be in the triangle, on the triangle, or outside of the triangle.

Notice that the perpendicular bisectors of the sides of the triangles do not necessarily pass through the vertices of the triangles.

 NOTE: The point of concurrency of the perpendicular bisectors of the sides of a triangle (the circumcenter) is the center of a circumscribed circle about the triangle.

A circumscribed circle is a circle around the outside of a figure passing through all of the vertices of the figure. In this case, passing through the three vertices of the triangle. Since the radii of the circle are congruent, a circumcenter is equidistant from vertices of the triangle.   In a right triangle, the perpendicular bisectors intersect ON the hypotenuse of the triangle. Since the center of the circumscribed circle lies on the hypotenuse, the hypotenuse becomes the diameter of the circle.  Locate the circumcenter through construction: We have seen how to construct perpendicular bisectors of the sides of a triangle. Simply construct the perpendicular bisectors for all three sides of the triangle. The point where they intersect is the circumcenter. Remember that the perpendicular bisectors of the sides of a triangle may not necessarily pass through the vertices of the triangle. Actually, finding the intersection of only 2 perpendicular bisectors will find the circumcenter. Finding the third perpendicular bisector, however, will ensure more accuracy of the find. Construct a circle circumscribed about a triangle. Steps: 1. locate the circumcenter by constructing the perpendicular bisectors of at least two sides of the triangle. 2. place compass point on the circumcenter and stretch to any one of the vertices. 3. draw circumscribed circle.  