In triangle ` A B C ,l e tR=c i r c u m r a d i u s ,r= i n r a d i u sdot` If `r` is the distance between the circumcenter and the incenter, the ratio `R/r` is equal to (a)`sqrt(2)-1` (b) `sqrt(3)-1` (c)`sqrt(2)+1` (d) `sqrt(3)+1`

Prove that the distance between the circumcenter and the incenter of triangle ABC is sqrt(R^2-2R r)

In triangle A B C , let /_c=pi/2dot If r is the inradius and R is circumradius of the triangle, then 2(r+R) is equal to a+b (b) b+c c+a (d) a+b+c

The locus of the mid-points of the chords of the circle of lines radiùs r which subtend an angle pi/4 at any point on the circumference of the circle is a concentric circle with radius equal to (a) (r)/(2) (b) (2r)/(3) (c) (r )/(sqrt(2)) (d) (r )/(sqrt(3))

In triangle ABC, if r_(1) = 2r_(2) = 3r_(3) , then b : c is equal to

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

Three equal circles each of radius r touch one another. The radius of the circle touching all the three given circles internally is (2+sqrt(3))r (b) ((2+sqrt(3)))/(sqrt(3))r ((2-sqrt(3)))/(sqrt(3))r (d) (2-sqrt(3))r

If in triangle A B C ,sumsin(A/2)=6/5a n dsumI I_1=9 (where I_1,I_2 a n d I_3 are excenters and I is incenter, then circumradius R is equal to (15)/8 (b) (15)/4 (c) (15)/2 (d) 4/(12)

The largest area of the trapezium inscribed in a semi-circle or radius R , if the lower base is on the diameter, is (a) (3sqrt(3))/4R^2 (b) (sqrt(3))/2R^2 (3sqrt(3))/8R^2 (d) R^2

Values (s)(-i)^(1/3) is/are (sqrt(3)-i)/2 b. (sqrt(3)+i)/2 c. (-sqrt(3)-i)/2 d. (-sqrt(3)+i)/2

If the incircle of the Delta ABC touches its sides at L , M and N and if x , y , z be thecircumradii of the triangles MIN , NIL and LIM respectively, where I is the incentre, then the product xyz is equal to: (A) R r^2 (B) r R^2 (C) 1/2R r^2 (D) 1/2r R^2