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

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

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_________

A triangle is inscribed in a circle of radius 1. The distance between the orthocentre and the circumcentre of the triangle cannot be

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 sides of triangle are in the ratio 3 : 5 : 7 , then prove that the minimum distance of the circumcentre from the side of triangle is half the circmradius

Tangents are drawn to the parabola at three distinct points. Prove that these tangent lines always make a triangle and that the locus of the orthocentre of the triangle is the directrix of the parabola.

Show that the line joining the incenter to the circumcenter of triangle A B C is inclined to the side B C at an angle tan^(-1)((cosB+cosC-1)/(sinC-sinB))

Let O be the circumcentre and H be the orthocentre of an acute angled triangle ABC. If A gt B gt C , then show that Ar (Delta BOH) = Ar (Delta AOH) + Ar (Delta COH)

ABC is an acute angled triangle with circumcenter O and orthocentre H. If AO=AH, then find the angle A.

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.