To prove:

Perpendicular bisectors of the sides of a triangle are concurrent.

Assuming:

ABC be a triangle with vertices A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃).

Let D, E and F be the midpoints of the sides BC, CA and AB, respectively.

Explanation:

Thus, the coordinates of D, E and F are

Let mD, mE and mF be the slopes of AD, BE and CF respectively.

∴ Slope of BC × mD = -1

⇒

⇒ mD

Thus, the equation of AD

⇒

⇒

⇒

Similarly, the respective equations of BE and CF are

Let L₁, L₂ and L₃ represent the lines (1), (2) and (3), respectively.
Adding all the three lines,

We observe:
1 ⋅ L₁ + 1⋅L₂ + 1⋅L₃ = 0

Hence proved, the perpendicular bisectors of the sides of a triangle are concurrent.