Question

If $a,b,c$ are sides of a scalene triangle, then the value of $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is always

• A. non-positive
• B. positive
• C. non-negative
• D. negative

Answer 1

Answer: (D)

$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=3abc-a^3-b^3-c^3$
$=-(a+b+c)\left(a^2+b^2+c^2-ab-bc-ca\right)$
$=-(a+b+c)\times \frac{1}{2}\times \left[(a-b{)}^2+(b-c{)}^2+(c-a{)}^2\right]$
$=$ always negative as $a,b,c$ are sides of the triangle.