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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     Similarly, the respective equations of BE and CF are  Let L1, L2 and L3 represent the lines (1), (2) and (3), respectively.

We observe:
1
L1 + 1L2 + 1L3 = 0

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

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