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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.

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  • $\begingroup$ I found out that $y = csin B$ similarly for other angles. Should I have to convert it into cos and put into the given relationship ? $\endgroup$ – Partha Sarathi Das Oct 16 '17 at 5:31
  • $\begingroup$ what are $x,y,z$ ? your definition is ambiguous. $\endgroup$ – DeepSea Oct 16 '17 at 5:32
  • $\begingroup$ x,y,z are the perpendiculars on each side of the triangle from vertices A,B,C respectively $\endgroup$ – Partha Sarathi Das Oct 16 '17 at 5:34
  • $\begingroup$ These perpendiculars are named "altitudes". $\endgroup$ – Jean Marie Oct 16 '17 at 7:03
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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

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  • $\begingroup$ I Thanks for the suggestion, the hint suggests the whole answer though @lab $\endgroup$ – Partha Sarathi Das Oct 16 '17 at 5:47

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