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Mohammed furqaan Hussain

Aug 23, 2013

# If a2+b2+c2-ab-bc-ca=0,prove that a=b=c.

If a2+b2+c2-ab-bc-ca=0,prove that  a=b=c

Kishore Kumar

Consider, a2 + b2 + c2 â€“ ab â€“ bc â€“ ca = 0 Multiply both sides with 2, we get 2( a2 + b2 + c2 â€“ ab â€“ bc â€“ ca) = 0 â‡’ 2a2 + 2b2 + 2c2 â€“ 2ab â€“ 2bc â€“ 2ca = 0 â‡’ (a2 â€“ 2ab + b2) + (b2 â€“ 2bc + c2) + (c2 â€“ 2ca + a2) = 0 â‡’ (a â€“b)2 + (b â€“ c)2 + (c â€“ a)2 = 0 Since the sum of square is zero then each term should be zero â‡’ (a â€“b)2 = 0,  (b â€“ c)2 = 0, (c â€“ a)2 = 0 â‡’ (a â€“b) = 0,  (b â€“ c) = 0, (c â€“ a) = 0 â‡’ a = b,  b = c, c = a âˆ´ a = b = c

Agam Gupta

Consider, a2 + b2 + c2 â€“ ab â€“ bc â€“ ca = 0 Multiply both sides with 2, we get 2( a2 + b2 + c2 â€“ ab â€“ bc â€“ ca) = 0 â‡’ 2a2 + 2b2 + 2c2 â€“ 2ab â€“ 2bc â€“ 2ca = 0 â‡’ (a2 â€“ 2ab + b2) + (b2 â€“ 2bc + c2) + (c2 â€“ 2ca + a2) = 0 â‡’ (a â€“b)2 + (b â€“ c)2 + (c â€“ a)2 = 0 Since the sum of square is zero then each term should be zero â‡’ (a â€“b)2 = 0,  (b â€“ c)2 = 0, (c â€“ a)2 = 0 â‡’ (a â€“b) = 0,  (b â€“ c) = 0, (c â€“ a) = 0 â‡’ a = b,  b = c, c = a âˆ´ a = b = c.
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