X=a/b+c Y=b/c+a Z=c/a+b求证 x/1+x + y/1+y + z/1+z=1
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![X=a/b+c Y=b/c+a Z=c/a+b求证 x/1+x + y/1+y + z/1+z=1](/uploads/image/z/13893392-56-2.jpg?t=X%3Da%2Fb%2Bc+Y%3Db%2Fc%2Ba+Z%3Dc%2Fa%2Bb%E6%B1%82%E8%AF%81+x%2F1%2Bx+%2B+y%2F1%2By+%2B+z%2F1%2Bz%3D1)
X=a/b+c Y=b/c+a Z=c/a+b求证 x/1+x + y/1+y + z/1+z=1
X=a/b+c Y=b/c+a Z=c/a+b
求证 x/1+x + y/1+y + z/1+z=1
X=a/b+c Y=b/c+a Z=c/a+b求证 x/1+x + y/1+y + z/1+z=1
∵x=a/(b+c),y=b/(c+a),z=c/(a+b).
∴以上各等式变形为:
1/x=(b+c)/a=〔(a+b+c)/a〕-1 ,
1/y=(c+a)/b=〔(a+b+c)/b〕-1 ,
1/z=(a+b)/c=〔(a+b+c)/c〕-1,(分数的拆分)
∴继续向所求变形:(1/x)+1=(1+x)/x=(a+b+c)/a,→x/(1+x)=a/(a+b+c).
同理:y/(1+y)=b/(a+b+c),z/(1+z)=c/(a+b+c).
∴x/(1+x)+y/(1+y)+z/(1+z)
=a/(a+b+c)+b/(a+b+c)+c/(a+b+c)
=(a+b+c)/(a+b+c)
=1.