Question

Using a2 + b2 + c2 - ab - ca =
{(a - b)? +
(b-c)2 + (c - a)?} Prove that if
a? + b2 + c = ab + bc + ca, then a=b= c.​

Answer 1

Answer: Proof given below

Step-by-step explanation:

a^2 + b^2 + c^2 - ab -bc- ca =

(a - b)^2+(b-c)^2 + (c - a)^2 ............. (1)

Given that a^2 + b^2 + c^2 = ab + bc + ca

So, a^2 + b^2 + c^2 -ab - bc - ca= 0

Putting in.. (1), we get

LHS of (1) become zero

0= (a - b)^2+(b-c)^2 + (c - a)^2

Since the sum of square is zero then each term should be zero

So, (a –b)2 = 0, (b – c)2 = 0, (c – a)2 = 0 So, (a –b) = 0, (b – c) = 0, (c – a) = 0

So, a = b, b = c, c = a

Hence, a = b = c