X/(Y+Z)+Y/(X+Z)+Z/(X+Y)=1,求X^2/(Y+Z)+Y^2/(X+Z)+Z^2/(X+Y)=

Question

X/(Y+Z)+Y/(X+Z)+Z/(X+Y)=1,求X^2/(Y+Z)+Y^2/(X+Z)+Z^2/(X+Y)=

Answer 1

X/(Y+Z)+Y/(X+Z)+Z/(X+Y)=1
设x/(y+z)=a,y/(x+z)=b,z/(x+y)=c
∴a+b+c=1
x=a(y+z)
y=b(x+z)
z=c(x+y)
相加：x+x+z=(b+c)x+(a+c)y+(a+b)z
∴[1-(b+c)]x+[1-(a+c)]y+[1-(a+b)]z=0
∴ax+by+cz=0
即X^2/(Y+Z)+Y^2/(X+Z)+Z^2/(X+Y)=0