Question

设x/y=a/b,求证((x^2+a^2)/(x+a))+((y^2+b^2)/(y+b))=((x-y)^2+(a+b)^2)/(x+y+a+b)

Answer 1

右边的（x-y）应该是（x+y）吧。
证明：令 x/y=a/b=k
则 x=ky,a=kb;
左边=((k^2y^2+k^2a^2)/(ky+kb))+((y^2+b^2)/(y+b))
=(k(y^2+b^2)/(y+b))+((y^2+b^2)/(y+b))
=((k+1)(y^2+b^2))/(y+b)
右边=（(ky+y)^2+(kb+b)^2)/(ky+y+kb+b)
=((k+1)^2y^2+(k+1)^2b^2)/((k+1)y+(k+1)b)
=((k+1)y^2+(k+1)b^2)/(y+b)
=((k+1)(y^2+b^2))/(y+b)
所以 左边=右边=((k+1)(y^2+b^2))/(y+b)