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If the angles A,B,C of a triangle are in...

If the angles A,B,C of a triangle are in A.P. and sides a,b,c, are in G.P., then prove that `a^2, b^2,c^2` are in A.P.

Text Solution

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Given `2B = A + C`
or `3B = pi " or " B = pi//3` (i)
Also a, b, c are in G.P. `rArr b^(2) = ac` (ii)
Now, `cos B = cos 60^(@) = (1)/(2) = (c^(2) a^(2) -b^(2))/(2ca)`
or `ca = c^(2) + a^(2) - b^(2)`
or `2b^(2) = c^(2) + a^(2)` [by using Eq. (ii)]
Hence, `a^(2), b^(2), c^(2)` are in A.P.
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