If bc+cd=2bd and a+c=2b, then which one of the following is correct?

Explanation

From a+c = 2b, substitute 2b into the first equation: c(b+d) = 2bd \implies c = \frac{2bd}{b+d}. This indicates b, c, d are in Harmonic Progression (HP) and a, b, c are in Arithmetic Progression (AP). Substituting c = 2b - a into c(b+d) = 2bd expands to 2b^2 - ab - ad = 0. Using the identity relationships, it simplifies perfectly to ad - bc = 0.

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