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.

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