(a) negative
Let x = b + c – a, y = c + a – b, z = a + b – c
Since a, b, c are the sides of a non-equilateral triangle, by the triangle inequality a + b > c, b + c > a, c + a > b
⇒ a + b – c > 0, b + c – a > 0, c + a – b > 0
⇒ z > 0, x > 0, y > 0
Also, x, y, z are distinct.
∴ AM > GM
\(c=\frac{x+y}{2}>\sqrt{xy}\)
\(b=\frac{x+z}{2}>\sqrt{xz}\)
\(a=\frac{y+z}{2}>\sqrt{yz}\)
∴ abc > \(\sqrt{xy}.\sqrt{xz}.\sqrt{yz}\)
⇒ abc > xyz ⇒ abc > (b + c – a) (c + a – b) (a + b – c)
⇒ (b + c – a) (c + a – b) (a + b – c) – abc < 0, i.e., negative.
