Prove that : `1/(r_1)+1/(r_2)+1/(r_3)=1/r`

Prove that : (r_1)/(b c)+(r_2)/(c a)+(r_3)/(a b)=1/r-1/(2R)

Prove that (r_1-r)(r_2-r)(r_3-r)=4R r^2

Prove that : (r_1+r_2) tan (C )/(2) = (r_3- r) cot ( C)/(2) = c

Let A B C be a triangle with incenter I and inradius rdot Let D ,E ,a n dF be the feet of the perpendiculars from I to the sides B C ,C A ,a n dA B , respectively. If r_1,r_2a n dr_3 are the radii of circles inscribed in the quadrilaterals A F I E ,B D I F ,a n dC E I D , respectively, prove that (r_1)/(r-1_1)+(r_2)/(r-r_2)+(r_3)/(r-r_3)=(r_1r_2r_3)/((r-r_1)(r-r_2)(r-r_3))

Prove that r_(1) r_(2) + r_(2) r_(3) + r_(3) r_(1) = (1)/(4) (a + b + c)^(2)

If r,r_(1) ,r_(2), r_(3) have their usual meanings , the value of 1/(r_(1))+1/(r_(2))+1/(r_(3)) , is

Prove that (r_(1) -r)/(a) + (r_(2) -r)/(b) = (c)/(r_(3))

If the lengths of the perpendiculars from the vertices of a triangle ABC on the opposite sides are p_(1), p_(2), p_(3) then prove that (1)/(p_(1)) + (1)/(p_(2)) + (1)/(p_(3)) = (1)/(r) = (1)/(r_(1)) + (1)/(r_(2)) + (1)/(r_(3)) .

Prove that (r_(1+r_2))/1=2R

prove that : triangle ABC, 1/r_1+1/r_2+1/r_3=1/r