Question

If $a\cos A=b\cos B,$ prove that $△ABC$ is either isosceles, or right angled.

Answer 1

Since $a\cos A=b\cos B$, we have

$k\sin A\cos A=k\sin B\cos B$

or $\sin 2A=\sin 2B$.

$\therefore 2A=2B$ or $\pi -2B$

$\therefore A=B$ i.e the $△ABC$ is isosceles

or $A+B=\pi /2$ so that $C=\pi /2$ i.e $△ABC$ is right angled at $C$.