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.