# The Perpendicular Bisectors of a Triangle are Concurrent

Here we prove that the right bisectors of a triangle are concurrent.

Let , and be the vertices of the triangle . Let , and be the midpoints of , and respectively. Since is the midpoint of , then

If is the slope of , then we use the two point formula to find the slope of line

Since the perpendicular bisector is perpendicular to the side , its slope is given by using the condition of a perpendicular slope:

The equation of perpendicular passing through with the slope is

For the equation of the perpendicular bisector , we just replace with , with and with in (iii) (i.e. ), so

For the equation of the perpendicular bisector , we just replace with , with and with in (iv) (i.e. ), so

To see whether the perpendicular bisector (iii), (iv) and (v) are concurrent, consider the determinant:

This shows that the perpendicular bisectors of the triangle are concurrent.