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

In Delta ABC it is given distance between the circumcentre (O) and orthocentre (H) is R sqrt(1-8 cos A cos B cos C) . If Q is the midopoint of OH, then AQ is

Prove that the area of a sector of circle with radius r and central angle theta^@ is 1/2 r^2 theta .

All the notations used in statemnt I and statement II are usual. Statement I: In triangle ABC, if (cos A)/(a )=(cos B)/(b)=(cos C)/(c). then value of (r_(1)+r_(2)+r_(3))/(r) is equal to 9. Statement II: In Delta ABC:(a)/(sin A) =(b)/(sin B) =(c)/(sin C)=2R, where R is circumradius.

If the lengths of the perpendiculars from the vertices of a triangle ABC on the opposite sides are p_(1), p_(2), p_(3) then prove that (1)/(p_(1)) + (1)/(p_(2)) + (1)/(p_(3)) = (1)/(r) = (1)/(r_(1)) + (1)/(r_(2)) + (1)/(r_(3)) .

In Delta ABC, a=5, b=4, c=3. G is the centroid of triangle. If R_(1) be the circum radius of triangle GAB then the value of (a)/(65) R_(1)^(2) must be

In Delta ABC , the ratio a/sin A=b/sinB=c/sinC is not always equal to (All symbols used have usual meaning in a triangle.) a. 2R, where R is the circumradius b. (abc)/(2Delta), where Delta is the area of the triangle c. 2/3 (a^(2)+ b^(2) +c^(2)) ^(1/2) d. ((abc)^(2/3))/((h_(1)h_(2) h_(3))^(1/3)

Prove that the equation p cos x - q sin x =r admits solution for x only if -sqrt(p^(2)+q^(2)) lt r lt sqrt(p^(2)+q^(2))

If in a triangle of base 'a', the ratio of the other two sides is r ( <1).Show that the altitude of the triangle is less than or equal to (ar)/(1-r^2)

Prove that the triangle ABC is equilateral if cotA+cotB+cotC=sqrt3

If R_(1) is the circumradius of the pedal triangle of a given triangle ABC, and R_(2) is the circumradius of the pedal triangle of the pedal triangle formed, and so on R_(3), R_(4) ..., then the value of underset( i=1)overset(oo)sum R_(i) , where R (circumradius) of DeltaABC is 5 is