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

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

Let A B C be a triangle having O and I as its circumcenter and incentre, respectively. If R and r are the circumradius and the inradius, respectively, then prove that (O I)^2=R^2-2R rdot Further show that the triangle B I O is a right angled triangle if and only if b is arithmetic mean of a and c.

Given that P(3,1),Q(6. 5), and R(x , y) are three points such that the angle P Q R is a right angle and the area of R Q P is 7, find the number of such points Rdot

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

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 radii r_1, r_2,r_3 of the escribed circles of the triangle A B C are in H.P. If the area of the triangle is 24 c m^2a n d its perimeter is 24cm, then the length of its largest side is 10 (b) 9 (c) 8 (d) none of these

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 greatest angle of the triangle is given by

Let A B C be a triangle right-angled at Aa n dS be its circumcircle. Let S_1 be the circle touching the lines A B and A C and the circle S internally. Further, let S_2 be the circle touching the lines A B and A C produced and the circle S externally. If r_1 and r_2 are the radii of the circles S_1 and S_2 , respectively, show that r_1r_2=4 area ( A B C)dot

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

In ABC with usual notations, if r=1,r_1=7 and R=3, the ABC is (a) equilateral (b) acute angled which is not equilateral (c)obtuse angled (d) right angled