Prove using vectors that the distance of the circumcenter of the `Delta ABC` form the centroid is `sqrt(R^(2) - (1)/(9)(a^(2) + b^(2) + c^(2)))` where R is the circumradius.

Prove that the distance of the incentre of Delta ABC from A is 4R sin B/2 sin C/2.

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

If in Delta ABC, 8R^(2) = a^(2) + b^(2) + c^(2) , then the triangle ABC is

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

In DeltaABC, angle C = 2 angle A, and AC = 2BC , then the value of (a^(2) + b^(2) c^(2))/(R^(2)) (where R is circumradius of triangle) is ______

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

(i) A variable plane, which remains at a constant distance '3p' from the origin cuts the co-ordinate axes at A, B, C. Show that the locus of the centroid of the triangle ABC is : (1)/(x^(2)) + (1)/(y^(2)) + (1)/(z^(2)) = (1)/(p^(2)) . (ii) A variable is at a constant distance 'p' from the origin and meets the axes in A, B, C respectively, then show that locus of the centroid of th triangle ABC is : (1)/(x^(2)) + (1)/(y^(2)) + (1)/(z^(2)) = (9)/(p^(2)).

Circumradius of a Delta ABC is 2, o is the circumradius of a Delta ABC is 2, o is the (1)/(64)(AH^(2)+BC^(2))(BH^(2)+AC^(2))(CH^(2)+AB^(2)) is equal to

In any Delta ABC, prove the following : r_(1)=r cot((B)/(2))cot((C)/(2))