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

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_2))/1=2R

Show that (r_(1)+ r _(2))(r _(2)+ r _(3)) (r_(3)+r_(1))=4Rs^(2)

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

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

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

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

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

Prove that r_(1)^(2)+r_(2)^(2) +r_(3)^(3) +r^(2) =16R^(2) -a^(2) -b^(2) -c^(2). where r= in radius, R = circumradius,, r_(1), r_(2), r_(3) are ex-radii.

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)))