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

In any triangle ABC, if 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 8R^2=a^2+b^2+c^2 then show that the triangle is right angled

If theta_1,theta_2, theta_3 radians be the angles subtended by the arcs of lengths l_1,l_2,l_3 at the centres of the circle whose radii are r_1,r_2,r_3 respectively then show that the angle subtended at the centre by the arc of length (l_1+l_2+l_3) of a circle whose radius is (r_1+r_2+r_3) will be (r_1theta_1+r_2theta_2+r_3theta_3)/(r_1+r_2+r_3) radian.

If r/(r_1)=(r_2)/(r_3), then

If in a triangle (1-r_1/r_2)(1-r_1/r_3)=2, then the triangle is

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))

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

If 5, 5r and 5r^(2) are the lengths of the sides of a triangle, then r cannot be equal to