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प्रश्न
Prove that medians of a triangle are concurrent
उत्तर
Consider ∆ABC.
Let P, Q, R be the midpoints of the sides BC, CA, AB respectively.
Let the medians BQ and CR intersect at G.
To prove that the third median AP also passes through G.
Let `bar"a", bar"b", bar"c", bar"p", bar"q", bar"r", bar"g"` be the position vectors of the points A, B, C, P, Q, R, G respectively.
Since P, Q, R are the mid-points of the sides BC, CA, AB respectively
∴ By midpoint formula, we get
`bar"p" = (bar"b" + bar"c")/2` .......(i)
`bar"q" = (bar"c" + bar"a")/2` .......(ii)
`bar"r" = (bar"a" + bar"b")/2` .......(iii)
From (i), (ii) and (iii), we get
`2bar"p" =bar"b" + bar"c" ⇒ 2bar"p" + bar"a" = bar"a" + bar"b" + bar"c"`
`2bar"q" = bar"c" + bar"a" ⇒ 2bar"q" + bar"b" = bar"a" + bar"b" + bar"c"`
`2bar"r" = bar"a" + bar"b" ⇒ 2bar"r" + bar"c" = bar"a" + bar"b" + bar"c"`
∴ `(2"p" + bar"a")/3 = (2bar"q" + bar"b")/3 = (2bar"r" + bar"c")/3 = (bar"a" + bar"b" + bar"c")/3`
∴ `(2"p" + bar"a")/(2 + 1) = (2bar"q" + bar"b")/(2 + 1) = (2bar"r" + bar"c")/(2 +1) = (bar"a" + bar"b" + bar"c")/3`
= `bar"g"` .......(say)
This shows that the point G whose position vector is `bar"g"` lies on the three medians AP, BQ, CR dividing them internally in the ratio 2:1.
Hence, the three medians are concurrent.
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