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In a $\triangle A B C$, if $\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$, then $\triangle A B C$ is
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Verified Answer
The correct answer is:
equilateral
$\text { In } \triangle A B C, \frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$
By sine rule,
$\begin{array}{lc}
\Rightarrow & \frac{\cos A}{k \sin A}=\frac{\cos B}{k \sin B}=\frac{\cos C}{k \sin C} \\
\Rightarrow & \cot A=\cot B=\cot C \\
\Rightarrow & A=B=C
\end{array}$
Hence, all angles are equal so the $\triangle A B C$ is an equilateral.
By sine rule,
$\begin{array}{lc}
\Rightarrow & \frac{\cos A}{k \sin A}=\frac{\cos B}{k \sin B}=\frac{\cos C}{k \sin C} \\
\Rightarrow & \cot A=\cot B=\cot C \\
\Rightarrow & A=B=C
\end{array}$
Hence, all angles are equal so the $\triangle A B C$ is an equilateral.
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