`(r_3+r_1)(r_3+r_2)sinC=2r_3sqrt(r_2r_3+r_3r_1+r_1r_2)`

r + r_3 + r_1 - r_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))

If r_(1), r_(2), r_(3) are radii of the escribed circles of a triangle ABC and r it the radius of its incircle, then the root(s) of the equation x^(2)-r(r_(1)r_(2)+r_(2)r_(3)+r_(3)r_(1))x+(r_(1)r_(2)r_(3)-1)=0 is/are :

(r_(2)+r_(3))sqrt((r r_(1))/(r_(2)r_(3)))=

Statement I In a Delta ABC, if a lt b lt c and r is inradius and r_(1) , r_(2) , r_(3) are the exradii opposite to angle A,B,C respectively, then r lt r_(1) lt r_(2) lt r_(3). Statement II For, DeltaABC r_(1)r_(2)+r_(2)r_(3)+r_(3)r_(1)=(r_(1)r_(2)r_(3))/(r)

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 the questions a(r r_(1) + r_(2)r_(3)) = b(r r_(2) + r_(3) r_(1)) = c (r r_(3) + r_(1) r_(2))

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

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