Perpendicular Bisector Point Of Concurrency

Proving the Concurrency of the Perpendicular Bisectors of a Triangle

By Sharon K. O�Kelley

Let�s show that the iii perpendicular bisectors of the sides of a triangle are concurrent which means that they intersect at one point.

To practice then, let�s consider triangle ABC equally synthetic in Figure i. In triangle ABC, perpendicular bisectors FD and FE have been constructed. If a line is a perpendicular bisector of the side of a triangle, then it bisects the side into two halves and forms correct angles with the side. Therefore, the following can exist determined from the effigy�

AD = DC and AE = EB

Angles FDA, FDC, FEA and February are each 90 degrees.

Figure 1

Note that in Figure 1, perpendicular bisectors FD and FE intersect at point F. To bear witness that the three perpendicular bisectors of triangle ABC are concurrent, we must show that the third perpendicular bisector goes through point F as well.

For purposes of convenience, perpendicular bisectors DF and FE have been shortened to segments FD and FE in Figure ii.

Figure 2

The Perpendicular Bisector Theorem states that if a point lies on the perpendicular bisector of a segment, it is equidistant from the endpoints of the bisected segment.

Hence, as Figure three shows, since bespeak F lies on perpendicular bisector FD, bespeak F is equidistant from points A and C; therefore, FA = FC. Since betoken F also lies on perpendicular bisector Iron, it is besides equidistant from points A and B; therefore, FA = FB. Using substitution, it tin can be ended that FA = FB = FC.

Figure iii

Since FC = FB, this means that signal F must be equidistant from points C and B equally well. Since points C and B are the endpoints of segment BC and point F is equidistant from those points, it tin can be concluded that point F lies on the perpendicular bisector of side BC. That perpendicular bisector has been synthetic as segment FG in Figure 4.

Figure four

Information technology can be concluded then that all iii perpendicular bisectors, FD, Atomic number 26, and FG, are concurrent at betoken F because indicate F is equidistant from all 3 vertices of the triangle. This point is too called the circumcenter considering information technology is the center of the circumvolve that circumscribes the triangle. In figure 5, the radii of the circumvolve are FA, FB, and FC.