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)

If in Delta ABC, (a -b) (s-c) = (b -c) (s-a) , prove that r_(1), r_(2), r_(3) are in A.P.

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)

In DeltaABC, R, r, r_(1), r_(2), r_(3) denote the circumradius, inradius, the exradii opposite to the vertices A,B, C respectively. Given that r_(1) :r_(2): r_(3) = 1: 2 : 3 The sides of the triangle are in the ratio

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+r_2)/(1+cosC)=2R

Let ABC be a triangle with incentre I and inradius r. Let D, E, F be the feet of the perpendiculars from I to the sides BC, CA and AB, respectively, If r_(2)" and "r_(3) are the radii of circles inscribed in the quadrilaterls AFIE, BDIF and CEID respectively, then prove that r_(1)/(r-r_(1))+r_(2)/(r-r_(2))+r_(3)/(r-r_(3))=(r_(1)r_(2)r_(3))/((r-r_(1))(r-r_(2))(r-r_(3)))

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

Prove that sum_(r=0)^n r(n-r)(.^nC_ r)^2=n^2(.^(2n-2)C_n)dot

If the polynomial f(x)=x^(2)+kx-1 be divided by (x - 2) and (x - 3), the respective remainders are r_(1)andr_(2) , then prove that 3r_(1)-2r_(2)+7=0 .