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Question

p, q and r are in AP, then prove that (p + 2q - r) (2q + r -p) (r + p - q) = 4pqr.

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Solution

Given p,q,r are in AP
qp=rq (distance between p & q is equal to distance between q & r)
2q=r+p......(1)
LHS=(p+2qr)(2q+rp)(r+pq)
=(p+r+pr)(r+p+rp)(r+pq) [ from eqn (1) sub for 2q]
=2p.2r(2qq) (from eqn (1))
=4pqr
=RHS
Hence proved

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