If the sides of a triangle are \( L_{r} \equiv x \cos \alpha_{r}+y \sin \alpha_{r}-p_{r}=0 \), \...
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0If the sides of a triangle are \( L_{r} \equiv x \cos \alpha_{r}+y \sin \alpha_{r}-p_{r}=0 \), \( (r=1,2,3) \) then Prove that
(i) Any line through the intersection of lines \( L_{1} \) and \( L_{2} \) and perpendicular to \( L_{3} \) is
\[
\frac{L_{1}}{\cos \alpha_{1} \cos \alpha_{3}+\sin \alpha_{1} \sin \alpha_{3}}=\frac{L_{2}}{\cos \alpha_{2} \cos \alpha_{3}+\sin \alpha_{2} \sin \alpha_{3}}
\]
(ii) the orthocentre is given by \( L_{1} \cos \left(\alpha_{2}-\alpha_{3}\right)=L_{2} \cos \left(\alpha_{3}\right. \) \( \left.-\alpha_{1}\right)=L_{3} \cos \left(\alpha_{1}-\alpha_{2}\right) \)
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