Prove that `(r+r_1)tan((B-C)/2)+(r+r_2)tan((C-A)/2)+(r+r_3)tan((A-B)/2)=0`

Prove that (r_1+r_2)/(1+cosC)=2R

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

Prove that r_1+r_2+r_3-r=4R

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)

If A + B + C = 180^(@) , prove that tan ""(A)/(2) tan ""(B)/(2) + tan ""(B)/(2) tan""( C)/(2) +tan""(C)/(2) tan"" (A)/(2) = 1

Prove that sum_(r=1)^(n) tan^(-1) ((2^(r -1))/(1 + 2^(2r -1))) = tan^(-1) (2^(n)) - (pi)/(4)

In any triangle A B C , prove that following : c/(a+b)=(1-tan(A/2)tan(B/2))/(1+tan(A/2)tan(B/2))

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 sum_(r=0)^(2n)r(.^(2n)C_r)^2=n^(4n)C_(2n) .

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