Each side of a triangle is multiplied with the sum of the squares of the other two sides. If the sum of all such possible results is 6 times the product of the sides, then the triangle must be
A
equilateral
B
isosceles
C
scalene
D
right angled
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A
equilateral
(a) equilateral Let $a, b, c$ be the lengths of the sides of the triangle. According to the question $\begin{array}{ll}& a\left(b^2+c^2\right)+b\left(c^2+a^2\right)+c\left(a^2+b^2\right)=6 a b c \\ \Rightarrow & a(b-c)^2+b(c-a)^2+c(a-b)^2=0 \\ \Rightarrow & a(b-c)=0, b(c-a)=0, c(a-b)=0 \quad\left[\because a(b-c)^2 \geq 0, b(-a)^2 \geq 0 \text { and } c(a-b)^2 \geq 0\right] \\ \Rightarrow & b=c, c=a \text { and } a=b \Rightarrow a=b=c\end{array}$ So, triangle $A B C$ is equilateral.
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The lengths of the sides of $\triangle\text{ABC}$ are consecutive integers. It $\triangle\text{ABC}$ has the same perimeter as an equilateral triangle with a side of length $9\ cm$, what is the length of the shortest side of $\triangle\text{ABC}?$