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Let f, g and h be the lengths of the per...

Let `f, g and h` be the lengths of the perpendiculars from the circumcenter of `Delta ABC` on the sides a, b, and c, respectively. Prove that `(a)/(f) + (b)/(g) + (c)/(h) = (1)/(4) (abc)/(fgh)`

Text Solution

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Distance of circumcenter from of side BC is R `cos A = f`
Similarly, `g = R cos B, h = R cos C`
`rArr (a)/(f) + (b)/(g) + (c)/(h) = (2R sin A)/(R cos A) + (2R sin B)/(R cos B) + (2R sin C)/(R cos C)`
`=2 (tan A + tan B + tan C)`
Also, `(a)/(f) (b)/(g) (c)/(h) = 8 tan A tan B tan C`
But in triangle, `tan A + tan B + tan C = tan A tan B tan C`. Thus,
`(a)/(f) + (b)/(g) + (c)/(h) = (1)/(4) (abc)/(fgh)`
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