If a, b, c are sides of a scalene triangle, then the value of |...

\begin{aligned} &\left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right|=\left|\begin{array}{ccc} a+b+c & a+b+c & a+b+c \\ b & c & a \\ c & a & b \end{array}\right| \\ &=(a+b+c)\left|\begin{array}{lll} 1 & 1 & 1 \\ b & c & a \\ c & a & b \end{array}\right| \\ &=(a+b+c)\left|\begin{array}{ccc} 0 & 0 & 1 \\ b-c & c-a & a \\ c-a & a-b & b \end{array}\right| \\ &=(a+b+c)\left[a b+b c+c a-a^2-b^2-c^2\right] \\ &=-(a+b+c)\left[(a-b)^2+(b-c)^2+(c-a)^2\right] \end{aligned}$ Since $a, b, c$ are sides of $a$ scalene triangle, therefore at least two of the $a, b, c$ will be unequal.\begin{array}{ll}\therefore \quad & (a-b)^2+(b-c)^2+(c-a)^2>0 \\ & \text { Also } a+b+c>0 \\ \therefore \quad & -(a+b+c)\left[(a-b)^2+\right. \\ & \left.(b-c)^2+(c-a)^2\right] < 0\end{array}$$