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 ⇒ y3−y2x3−x2×mD=−1 ⇒ mD=x3−x2y3−y2 Thus, the equation of AD y−y2−y32=−x3−x2y3−y2(x−x2+x32) ⇒ y=y2−y32=−x3−x2y3−y2(x−x2+x32) ⇒ 2y(y3−y2)−(y23−y22)=2x(x3−x2)+x23−x22 ⇒ 2x(x3−x2)+2y(y3−y2)−(x23−x22)−(y23−y22)=0          ...(i) Similarly, the respective equations of BE and CF are 2x(x1−x3)+2y(y1−y3)−(x21−x23)−(y21−y23)=0              ...(ii) 2x(x2−x1)+2y(y2−y1)−(x22−x21)−(y22−y21)=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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