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Prove using vectors: Medians of a triang...

Prove using vectors: Medians of a triangle are concurrent.

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Let `bar(a),bar(b),bar(c)` be the position vectors of the vertices A,B,C of `triangleABC and bar(d),bar(e),bar(f)` be the position vectors of the midpoints D,E,F of the sides BC, CA and AB respectivey.
Then by the midpoint formula,
`bar(d)=(bar(b)+bar(c))/(2),bar(e)=(bar(c)+bar(a))/(2),bar(f)=(bar(a)+bar(b))/(2)`
`:.2bar(d)=bar(b)+bar(c)," "2bar(e)=bar(c)+bar(a)," "2bar(f)=bar(a)+bar(b)`
`:.2bar(d)+bar(a)=bar(a)+bar(b)+bar(c)`
`2bar(e)+bar(b)=bar(a)+bar(b)+bar(c)`
`2bar(f)+bar(c)=bar(a)+bar(b)+bar(c)`
`:.(2bar(d)+bar(a))/(2+1)=(2bar(e)+bar(b))/(2+1)=(2bar(f)+bar(c))/(2+1)=(bar(a)+bar(b)+bar(c))/(3)=bar(g)` . . . (Say)
This show that the points G whose position vector is `bar(g)`, lies on the three medians AD,BE and CF dividing each of them internally in the ratio 2:1. Hence, the medians are concurrent in the point G and its position vector is `(bar(a)+bar(b)+bar(c))//3`.
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