Let the incircle with center I of ` A B C` touch sides BC, CA and AB at D, E, F, respectively. Let a circle is drawn touching ID, IF and incircle of ` A B C` having radius `r_2dot` similarly `r_1a n dr_3` are defined. Prove that `(r_1)/(r-r_1)dot(r_2)/(r-r_2)dot(r_3)/(r-r_3)=(a+b+c)/(8R)`

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

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

Incircle of A B C touches the sides BC, CA and AB at D, E and F, respectively. Let r_1 be the radius of incircle of B D Fdot Then prove that r_1=1/2((s-b)sinB)/((1+sin(B/2)))

Prove that : (r_1)/((s-b)(s-c))+(r_2)/((s-c)(s-a))+(r_3)/((s-a)(s-b))=3/r

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

In a triangle ABC prove that a(rr_1+r_2r_3)=b(rr_2+r_3r_1)=c(rr_3+r_1r_2)=abc

In Figure, the incircle of A B C touches the sides B C ,\ C A and A B\ a t\ D ,\ E and F respectively. Show that A F+B D+C E=A E+B F+C D=1/2(P e r i m e t e r\ of\ A B C) .

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

Prove that (r+r1)tan((B-C)/2)+(r+r2)tan((C-A)/2)+(r+r3)tan((A-B)/2)=0

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))