Prove that the perpendicular bisectors of a triangle are concurrent.

Hello student,

Suppose D, E, F are the mid-points of sides BC,
CA, AB of ΔABC.
Also, suppose that the perpendicular bisectors of
sides BC and CA intersect each other in the point O.
Then, to prove that the perpendicular bisectors of
sides of a Δ are concurrent, it is sufficient to prove
that OF ⊥ AB.
Taking O as the Origin of Reference, position vectors of
A, B, ... are OA↑ = a↑, OB↑ = b↑, ... .
∴ by Mid-Point Formula,
d↑ = (b↑ + c↑)/2, e↑ = (c↑ + a↑)/2, ƒ↑ = (a↑ + b↑)/2. ... (1)
Since OD ⊥ BC, we have OD↑ ⊥ BC↑ so that their
dot product is 0, i.e.,
(OD↑) • (BC↑) = 0 ∴ (d↑) • (c↑ - b↑) = 0 ∴ [ (c↑ + b↑ )/2 ] • (c↑ - b↑ ) = 0
∴ (c↑)² - (b↑)² = 0 ...... (2)
Similarly, considering OE↑ ⊥ CA↑, we can show that
(a↑)² - (c↑)² = 0 ...... (3)
Adding (2) and (3), we get : (a↑)² - (b↑)² = 0
∴ (b↑)² - (a↑)² = 0
∴ (b↑ + a↑) • (b↑ - a↑) = 0
∴ [ (b↑ + a↑)/2 ] • (b↑ - a↑) = 0
∴ ƒ↑ • (b↑ - a↑) = 0
∴ OF↑ ⊥ AB↑

Thanks and Regards

Shaik Aasif

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