If `r_1=r_2+r_3+r` prove that the triangle is right angled .

If in a triangle r_1=r_2+r_3+r , prove that the triangle is right angled.

In a triangle ABC, 8R^2=a^2+b^2+c^2 prove that the triangle is right angled

Let ABC be a triangle having O and I as its circumcentre and incentre, respectively. If R and r are the circumradius and the inradius respectively, then prove that (IO) 2 =R 2 −2Rr. Further show that the triangle BIO is right angled triangle if and only if b is the arithmetic mean of a and c.

If in a triangle (1-(r_1)/(r_2))(1-(r_1)/(r_3))=2 then the triangle is right angled (b) isosceles equilateral (d) none of these

If 8R^(2)=a^(2) +b^(2) +c^(2). then prove that the Delta is right angled.

If r_(1) +r_(2)+ r = r_(3), then show that Delta is right angled.

If in a triangle ABC, r_(1)+r_(2)+r_(3)=9r , then the triangle is necessarily

In A B C , if r\ : r_1: R=2\ : 12\ :5, where all symbols have their usual meaning, then A B C is an acute angled triangle A B C is an obtuse angled triangle A B C is right angled which is not isosceles A B C is isosceles which is not right angled

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))

The conbined resistance R of two resistors R,& R_(2)(R_(1),R_(2) gt 0) is given by . (1)/(R)=(1)/(R_(1)) +(1)/(R_(2)). If R_(1)+R_(2)= constant Prove that the maximum resistance R is obtained by choosing R_(1)=R_(2)