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 , so
If is the slope of , then using two point formula to find slope of line
Since the perpendicular bisector is perpendicular to the side , so its slope is given as using condition of perpendicular slope is
Equation of perpendicular passing through with slope is
For the equation of perpendicular bisector , we just replace by , by and by in (iii) (i.e. ), so
For the equation of perpendicular bisector , we just replace by , by and by 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.