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_2a 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+r_2+r_3-r=4R

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

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

Prove that (r+r_1)tan((B-C)/2)+(r+r_2)tan((C-A)/2)+(r+r_3)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))

Prove that ""^(n-1)P_r+r^(n-1)p_r-1=^nP_r dot""

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

Prove that (3!)/(2(n+3))=sum_(r=0)^n(-1)^r((^n C_r)/(^(r+3)C_r))

If I is the incenter of Delta ABC and R_(1), R_(2), and R_(3) are, respectively, the radii of the circumcircle of the triangle IBC, ICA, and IAB, then prove that R_(1) R_(2) R_(3) = 2r R^(2)

Prove that sum_(r=1)^n(-1)^(r-1)(1+1/2+1/3++1/r)^n C_r=1/n .