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.`

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

A B C is an equilateral triangle of side 4c mdot If R ,ra n dh are the circumradius, inradius, and altitude, respectively, then (R+h)/h is equal to 4 (b) 2 (c) 1 (d) 3

The perimeter of a triangle ABC right angled at C is 70 and the inradius is 6, then |a-b|=

In triangle ABC if 2sin^(2)C=2+cos2A+cos2B , then prove that triangle is right angled.

in a triangle ABC , let angle C = (pi)/(2) if r the in radius and R is the circumradius of the triangle ABC , then 2( r+R) equals

The distance of incentre of the right-angled triangle ABC (right angled at A) from B and C are sqrt10 and sqrt5 , respectively. The perimeter of the triangle is _____

In triangle A B C , the equation of side B C is x-y=0. The circumcenter and orthocentre of triangle are (2, 3) and (5, 8), respectively. The equation of the circumcirle of the triangle is

If I is the incenter of Delta ABC and R_(1), R_(2), and R_(3) are, respectively, the radii of the circumcircle of the triangle IBC, ICA, and IAB, then prove that R_(1) R_(2) R_(3) = 2r R^(2)

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

O P Q R is a square and M ,N are the midpoints of the sides P Q and Q R , respectively. If the ratio of the area of the square to that of triangle O M N is lambda:6, then lambda/4 is equal to (a)2 (b) 4 (c) 2 (d) 16