If #a/(b+c) + b/(c+a) + c/(a+b)=1# prove that
#a^2/(b+c) + b^2/(c+a) + c^2/(a+b)=0#

Question

If #a/(b+c) + b/(c+a) + c/(a+b)=1# prove that
#a^2/(b+c) + b^2/(c+a) + c^2/(a+b)=0#

Savannah Anderson · Accepted Answer

See below.

We are aware of that #a/(b+c) + b/(c+a) + c/(a+b)-1=0# which is equivalent to #(a^3 + b^3 + a b c + c^3)/((a + b) (a + c) (b + c))=0# also #a^2/(b+c) + b^2/(c+a) + c^2/(a+b)=# #=(a^4 + a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + a b^2 c + b^3 c + a b c^2 + a c^3 + b c^3 + c^4)/((a + b) (a + c) (b + c))=0# but #a^4 + a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + a b^2 c + b^3 c + a b c^2 + a c^3 + b c^3 + c^4=(a+b+c)(a^3 + b^3 + a b c + c^3)# so #a^4 + a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + a b^2 c + b^3 c + a b c^2 + a c^3 + b c^3 + c^4=0# and so on #a^2/(b+c) + b^2/(c+a) + c^2/(a+b)=0#

Abigail Allen · Answer

Given the relationship

#a/(b+c)+b/(c+a)+c/(a+b)=1#

Multiplying both sides by #a+b+c# we get #(a+b+c!=0)#

#=>(a(a+b+c))/(b+c)+(b(a+b+c))/(c+a)+(c(a+b+c))/(a+b)=(a+b+c)#

#=>a^2/(b+c)+(a(b+c))/(b+c)+b^2/(c+a)+(b(c+a))/(c+a)+c^2/(a+b)+(c(a+b))/(a+b)=(a+b+c)#

#=>a^2/(b+c)+a+b^2/(c+a)+b+c^2/(a+b)+c=(a+b+c)#

#=>a^2/(b+c)+b^2/(c+a)+c^2/(a+b)=0#

Proved

Jace Anderson · Answer

undefined Given $ \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} = 1 $, we can prove that $ \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} = 0 $.

Starting with the given equation, we multiply both sides by $ a + b + c $ to clear the denominators:

$$ (a + b + c) \left( \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \right) = (a + b + c) \cdot 1 $$

$$ a + b + c = a + b + c $$

Now, let's rewrite the expression we need to prove:

$$ \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} = 0 $$

Multiply both sides by $ a + b + c $:

$$ (a + b + c) \left( \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} \right) = (a + b + c) \cdot 0 $$

$$ a^2 + b^2 + c^2 = 0 $$

Since $ a^2 + b^2 + c^2 $ cannot be equal to zero unless $ a = b = c = 0 $, which contradicts the given condition, the statement is proved.

If #a/(b+c) + b/(c+a) + c/(a+b)=1# prove that #a^2/(b+c) + b^2/(c+a) + c^2/(a+b)=0# ?

If #a/(b+c) + b/(c+a) + c/(a+b)=1# prove that #a^2/(b+c) + b^2/(c+a) + c^2/(a+b)=0#

If #a/(b+c) + b/(c+a) + c/(a+b)=1# prove that
#a^2/(b+c) + b^2/(c+a) + c^2/(a+b)=0#