Question

If $\frac{a}{\cos A}=\frac{b}{\cos B}=\frac{c}{\cos C}$, then show that $△ABC$ is equilateral

Answer 1

$\frac{a}{\cos A}=\frac{b}{\cos B}=\frac{c}{\cos C}=k$ (say)

$a=k\cos A,b=k\cos B,c=k\cos C$.........(i)

We know that

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$ (sine rule)

$a=2R\sin A,b=2R\sin B,c=2R\sin C$ .......... (ii)

From (i) and (ii)

$k\cos A=2R\sin A,k\cos B=2R\sin B,k\cos C=2R\sin C$

$\tan A=\frac{k}{2R},\tan B=\frac{k}{2R},\tan C=\frac{k}{2R}$

$\tan A=\tan B=\tan C$

$A=B=C$

$\Delta ABC$ is equilateral triangle.