In a triangle $\Delta ABC$ there are three perpendiculars $x,y,z$ from vertices $A,B,C$ respectively. Prove that $\cos A/x + \cos B/y + \cos C/z = 1/R$, where R is circumcentre of the triangle.
I can't find any suitable relationship between $\cos$ of all angles and the heights of the triangle. Any clue regarding this will be helpful.
Hint:
$$\dfrac{\cos B}y=\dfrac{\cos B}{c\sin A}=-\dfrac{\cos(A+C)}{2R\sin C\sin A}=\dfrac{1-\cot A\cot C}{2R}$$
Now for $A+B+C=\pi,$
$$\cot A\cot B+\cot B\cot C+\cot C\cot A=1$$ Proof