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Question

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


Solution

Let 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.

Thus, the coordinates of D, E and F are

D(x2+x32,y2+y32)

E(x1+x32,y1+y32) and

F(x1+x22,y1+y22)

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

Slope of BC×mD=1

 y3y2x3x2×mD=1

 mD=x3x2y3y2

Thus, the equation of AD

yy2y32=x3x2y3y2(xx2+x32)

 y=y2y32=x3x2y3y2(xx2+x32)

 2y(y3y2)(y23y22)=2x(x3x2)+x23x22

 2x(x3x2)+2y(y3y2)(x23x22)(y23y22)=0          ...(i)

Similarly, the respective equations of BE and CF are

2x(x1x3)+2y(y1y3)(x21x23)(y21y23)=0              ...(ii)

2x(x2x1)+2y(y2y1)(x22x21)(y22y21)=0              ...(iii)

Let L1, L2 and L3 respresent the lines (i), (ii) and (iii), respectively.

Adding all the three lines.

We observe:

1lL1+1.L2+1.L3=0

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

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