If `a/(b+c)=b/(c+a)=c/(a+b)`and `a+b+cne0` then show that a=b=c

if (a)/(b + c) = (b)/(c + a) = (c)/(a + b) and a + b + c ne 0 , show that each given ratio is equal to (1)/(2) .

if (a)/(b + c) = (b)/(c + a) = (c)/(a + b) and a + b + c = 0 : show that each given ratio is equal to -1 .

If (b+c)/a,(c+a)/b,(a+b)/c are in A.P. show that 1/a,1/b1/c are also in A.P. (a +b+c != 0) .

If a^2(b+c),b^2(c+a),c^2(a+b), are in A.P. show that either a ,b ,c are in A.P., or a b+b c+c a=0.

If a+b+c!=0 and |a b c b c a c a b|=0 , then prove that a=b=cdot

If a+b+c!= and |(a,b,c),(b,c,a),(c,a,b)|=0 then prove that a=b=c

If the roots of the equation a(b-c)x^2+b(c-a)x+c(a-b)=0 are equal, show that 2//b=1//a+1// c dot

If a, b, c are in AP amd b, c, a are in GP , then show that c, a, b are in H.P. Find a : b : c .

If a'=(bxxc)/(["a b c"]),b'=(cxxa)/(["a b c"]),c'=(axxb)/(["a b c"]) then show that axxa'+bxxb'+cxxc'=0 , where a,b and c are non-coplanar.

If (a+b x)/(a-b x)=(b+c x)/(b-c x)=(c+dx)/(c-dx)(x!=0), then show that a, b, c and d are in G.P.