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

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

The distance between the circumcenter and the orthocentre of the triangle whose vertices are (0,0),(6,8), and (-4,3) is Ldot Then the value of 2/(sqrt(5))L is_________

Consider a triangle PQR with coordinates of its vertices as P(-8,5), Q(-15, -19), and R (1, -7). The bisector of the interior angle of P has the equation which can be written in the form ax+2y+c=0. The distance between the orthocenter and the circumcenter of triangle PQR is

If the distance between incenter and one of the excenter of an equilateral triangle is 4 units, then find the inradius of the triangle.

Prove that the circumcenter, orthocentre, incenter, and centroid of the triangle formed by the points A(-1,11),B(-9,-8), and C(15 ,-2) are collinear, without actually finding any of them.

In Delta ABC if the orthocentre is (1,2) and the circumcenter is (0,0) then centroid of Delta ABC is.

In triangle ABC, the line joining the circumcenter and incenter is parallel to side BC, then cosA+cosC is equal to -1 (b) 1 (c) -2 (d) 2

The radii of two circles are R and r unit (R gtr). If the distance between the two centres of the circles be d unit, then prove that the length of their transversal common tangent =sqrt(d^(2) -(R+r)^(2)) unit.

If a plane meets the coordinate axes at A, B, C such that the centroid of the triangle ABC is the point (1,r,r^(2)) , then show that the equation of the plane is r^(2)x+ry+z=3r^(2) .

In an equilateral triangle , R:r:r_2 is equal to