If in `Delta ABC`, the distance of the vertices from the orthocenter are x,y, and z then prove that `(a)/(x) + (b)/(y) + (c)/(z) = (abc)/(xyz)`

The perpendicular from the vertices A, B and C of a triangle ABC on the opposite sides meet at O. If OA=x, OB=y and OC=z, prove that, a/x + b/y + c/z = (abc)/(xyz) .

If x , ya n dz are the distances of incenter from the vertices of the triangle A B C , respectively, then prove that (a b c)/(x y z)=cot(A/2)cot(B/2)cot(C/2)

If a^(x)=b^(y)=c^(z) and b^(2)=ac prove that (1)/(x)+(1)/(z)=(2)/(y)

If the lengths of the sides vec(BC), vec(CA) and vec(AB) of the triangle ABC be a:b:c respectively and the lengths of perpendicular from the circumcenter upon vec(BC), vec(CA) and vec(AB) be x, y, z respectively, then prove that, a/x + b/y + c/z = (abc)/(4xyz) .

If x + y + z = 0 , then prove that x^(3)+y^(3)+z^(3)= 3xyz

Perpendicular drawn from the vertices A,B,C upon opposite sides of triangle ABC passes through a fixed point O.if bar(OA)=x.bar(OB)=y and bar(OC)=z then prove that a/x+b/y+c/z=(abc)/(xyz)

Eliminate x , y and z from the following equation: (bx)/(y+z) =a ,(cy)/(z+x)=b,(az)/(x+y)=c

The coordiantes of the centroid of the triangle fromed by the points (x-y,y-z),(-x,-y)and(y,z) are -

If a, b, c and x, y, z are positive, then show that (a/x + b/y + c/z) (x/a + y/b + z/c) ge 9

Answer any one question : If x : a = y : b = z : c, then show that (x^3)/(a^3) + (y^3)/(b^3) + (z^3)/(c^3) = (3xyz)/(abc) .