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

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

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

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

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

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

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

Prove that (r_(2) + r_(3))/(1 + cos A) = (r_(3) + r_(1))/(1 + cos B) = (r_(1) + r_(2))/(1+ cos C)

In a triangle ABC, prove that r^2+r_1^2+r_2^2+r_3^2=16R^2-a^2-b^2-c^2.

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)