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

Prove that the distance between the circumcenter and the orthocentre of triangle ABC is Rsqrt(1-8cosAcosBcosC)

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

Given an isoceles triangle with equal side of length b and angle alpha lt pi//4 , then the distance between circumcenter O and incenter I is

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

Find the distance between the following pairs of poins. R(-3a, a), S(a, - 2a)

P is a point on the line y+2x=1, and Q and R two points on the line 3y+6x=6 such that triangle P Q R is an equilateral triangle. The length of the side of the triangle is (a) 2/(sqrt(5)) (b) 3/(sqrt(5)) (c) 4/(sqrt(5)) (d) none of these