The Perpendicular Bisectors of a Triangle are Concurrent

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

Let $A\left( {{x_1},{y_1}} \right)$, $B\left( {{x_2},{y_2}} \right)$ and $C\left( {{x_3},{y_3}} \right)$ be the vertices of the triangle $ABC$. Let $D$, $E$ and $F$ be the midpoints of $AB$, $BC$ and $CA$ respectively. Since $D$ is the midpoint of $AB$, then

If ${m_1}$ is the slope of $AB$, then we use the two point formula to find the slope of line

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

The equation of perpendicular $OD$ passing through $D$ with the slope $m$ is

For the equation of the perpendicular bisector $OE$, we just replace ${x_1}$ with ${x_2}$, ${x_2}$ with ${x_3}$ and ${x_3}$ with ${x_1}$ in (iii) (i.e. ${x_1} \to {x_2},\,\,{x_2} \to {x_3},\,\,{x_3} \to {x_1}$), so

For the equation of the perpendicular bisector $OF$, we just replace ${x_1}$ with ${x_2}$, ${x_2}$ with ${x_3}$ and ${x_3}$ with ${x_1}$ in (iv) (i.e. ${x_1} \to {x_2},\,\,{x_2} \to {x_3},\,\,{x_3} \to {x_1}$), 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.