In a triangle ABC if the angles A, B, C are in AP, then which one of the following is correct ?

c = a + b

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c^{2} = a^{2} + b^{2} - a b

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a^{2} = b^{2} + c^{2} - b c

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b^{2} = a^{2} + c^{2} - a c

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Solution

The correct option is D $b^{2} = a^{2} + c^{2} - a c$
Let the angles be $a, a + d, a + 2 d$ . Then
$a + a + d + a + 2 d = 180^{o}$
$a + d = 60^{o}$
So, $∠ B = 60^{o}$ ,
By cosine formula,
$cos (B) = \frac{a^{2} + c^{2} - b^{2}}{2 a c}$

$\Rightarrow \frac{1}{2} = \frac{a^{2} + c^{2} - b^{2}}{2 a c}$

$\Rightarrow b^{2} = a^{2} + c^{2} - a c$
Hence, D is correct.

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