1.

If in a triangle`A B C ,``(2cosA)/a+(cos B)/b+(2cosC)/c=a/(b c)+b/(c a)`, then prove that the triangle is right angled.

Answer» `(2cosA)/a+(cosB)/b +(2cosC)/c = a/(bc)+b/(ca)`
`=>2[cosA/a+cosC/c]+cosB/b = a/(bc)+b/(ca)`
`=>2[(c cosA+ acosC)/(ac)]+cosB/b = a/(bc)+b/(ca)`
From projection formula,
`acosA + c cosA = b`
So, our equation becomes,
`(2b)/(ac)+cosB/b = a/(bc)+b/(ca)`
`=>cosB/b = a/(bc)-b/(ca)`
`=>(a^2+c^2-b^2)/(2abc) = (a^2-b^2)/(abc)`
`=>a^2+c^2-b^2 = 2a^2-2b^2`
`=>c^2-a^2+b^2 = 0`
`=>b^2+c^2 = a^2`
It means, it satisfies pythagoras theorem, so it is a right angle triangle.


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