Question

Show that the perpendicular bisectors of the sides of a triangle are concurrent.

Answer 1

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

⇒ \(\frac{y_2-y_2}{x_2-x_2}\)  x mD = -1

⇒mD = \(-\frac{x_2-x_2}{y_2-y_2}\) 

Thus, the equation of AD

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.