In an acute angle triange ABC, a semicircle with radius `r_(a)` is constructed with its base on BC and tangent to the other two sides `r_(b) and r_(c)` are defined similarly. If r is the radius of the incircle of triangle ABC then prove that `(2)/(r) = (1)/(r_(a)) + (1)/(r_(b)) + (1)/(r_(c))`

Explanation

As shown in the figure, semicircle with its base on BC touches AB and AC at E and D respectively.

Now, area of `DeltaABC = " Area of " Delta ABI_(1) + " Area of " Delta ACI_(1)`
`:. Delta = (1)/(2) cr_(a) + (1)/(2) br_(a)`
`:. (1)/(r_(a)) = (c+b)/(2Delta)`...(1)
Similarly, `(1)/(r_(b)) = (c + a)/(2Delta)`...(2)
and `(1)/(r_(c)) = (a + b)/(2Delta)`..(3)
On adding Eqs. (1), (2) and (3), we get
`(1)/(r_(a)) + (1)/(r_(b)) + (1)/(r_(c)) = (1)/(2Delta) (b + c + c + a + a + b)`
`=(a + b + c)/(Delta)`
`(2s)/(rs)`
`= (2)/(r)`