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By vector method prove that the medians of a triangle are concurrent. - Mathematics and Statistics

By vector method prove that the medians of a triangle are concurrent.

Using vector method prove that the medians of a triangle are concurrent.

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Solution

Let A, B and C be the vertices of a triangle.
Let D, E and F be the midpoints of the sides BC, AC and AB respectively.

Let `bar(GA) = bara,bar(GB) = barb,bar(GC) = barc,bar(GD) = bard,bar(GE) =bar e and bar(GF) =bar f ` be position vectors
of points A, B, C, D, E and F respectively.
Therefore, by midpoint formula,

`bard=(barb+barc)/2,bare=(bara+barc)/2 and barf=(bara+barb)/2`

`2bard=barb+barc,2bare=bara+barc and 2bar f=bara+barb`

`2bard+bara=bara+barb+barc ,2bare+barb=bara+barb+barc and 2barf+barc=bara+barb+barc`

`(2bard+bara)/3=(2bare+barb)/3=(2barf+barc)/3=(bara+barb+barc)/3`

`Let " " barg=(bara+barb+barc)/3`

`therefore " We have " barg=(bara+barb+barc)/3=((2)bard+(1)bara)/3=((2)bare+(1)barb)/3=((2)barf+(1)barc)/3`

If G is the point whose position vector is  `barg` , then from the above equation it is clear that the point G lies on the medians AD, BE, CF
and it divides each of the medians AD, BE, CF internally in the ratio 2:1. Therefore, three medians are concurrent.

 

Concept: Vectors - Medians of a Triangle Are Concurrent
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