If `a^2+b^2+c^2=x^2+y^2+z^2=1,` then show that `a x+b y+c z leq1.`

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

If x^(2) : (by + cz) =y^(2) : (cz + ax) = z^(2) : (ax + by) = 1 , then show that a/(a+x) + b/(b+y) + c/(c+z) = 1 .

If a^2+b^2+c^2 =16, x^2+y^2+z^2 = 25 and ax+by+cz = 20 , then show that (a+b+c) prop (x+y+z)

If x^2/(by+cz)=y^2/(cz+ax)=z^2/(ax+by)=1 then show that a/(a+x)+b/(b+y)+c/(c+z)=1

If x/(b + c -a) = y/(c + a - b) = z/(a + b -c) , then show that (b -c) x + (c-a) y + (a -b) z = 0 .

If a x_1^2+b y_1^2+c z_1^2=a x_2 ^2+b y_2 ^2+c z_2 ^2=a x_3 ^2+b y_3 ^2+c z_3 ^2=d ,a x_2 x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that |(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3,y_3,z_3)|=(d-f){((d+2f))/(a b c)}^(1//2)

If (b+c-a)x = (c+a-b)y = (a+b-c) z =2,then show that (i/x+1/y) (1/y+1/z) (1/z+1/x) =abc.

If x : a = y : b = z : c , then prove that (a^(2) + b^(2) + c^(2))(x^(2) + y^(2) + z^(2)) = (ax + by + cz)^(2)

If a tan x+ b tan y+ c tan z=m , then show that the minimum value of tan^(2)x+tan^(2)y+tan^(2)z "is" (m^(2))/(a^(2)+b^(2)+c^(2)) .

If x/(b +c) = y/(c+a) = z/(a + b) , then show that a/(y+z-x) = b/(z+x - y) = c/(x + y - z) .