Answer :

To prove:

Perpendicular bisectors of the sides of a triangle are concurrent.


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

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

Slope of BC × mD = -1


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:
L1 + 1L2 + 1L3 = 0

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

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