Question

已知x^2/(y+z)+y^2/(z+x)+z^2/(x+y)=0 求x/(y+z)+y/(z+x)+z/(x+y)的值

Answer 1

设x/(y+z)+y/(z+x)+z/(x+y)=k
若x+y+z=0 x/(y+z)+y/(z+x)+z/(x+y)=-3恒成立
若x+y+z不为0 x/(y+z)+y/(z+x)+z/(x+y)=k同乘（x+y+z）
x（x+y+z）/(y+z)+y（x+y+z）/(z+x)+z（x+y+z）/(x+y)=k（x+y+z）
即(x^2+x(y+z))/(y+z)+(y^2+(z+x)/(z+x))+z^2+z(x+y)/(x+y)=k（x+y+z)
x^2/(y+z)+y^2/(z+x)+z^2/(x+y)+x(y+z))/(y+z)+y(z+x)/(z+x))+z(x+y)/(x+y)=k（x+y+z)
即x(y+z))/(y+z)+y(z+x)/(z+x))+z(x+y)/(x+y)=k（x+y+z)
k=1