Доказать, что если: x = a-b/a+b; y = b-c/b+c; z = c-a/c+a, то (1 + x)(1 + y)(1 + z) = (1 - x)(1 - y)(1 - z). Created by Dima4000. matematika-ru

c+a, то (1 + x)(1 + y)(1 + z) = (1 - x)(1 - y)(1

$(1+x)(1+y)(1+z)=(1+\frac{a-b}{a+b})(1+\frac{b-c}{b+c})(1+\frac{c-a}{c+a})=\\\\=\frac{a+b+a-b}{a+b}\cdot \frac{b+c+b-c}{b+c}\cdot \frac{c+a+c-a}{c+a}=\frac{2a}{a+b}\cdot \frac{2b}{b+c}\cdot \frac{2c}{c+a}=\frac{8\, abc}{(a+b)(b+c)(c+a)}\\\\\\(1-x)(1-y)(1-z)=(1-\frac{a-b}{a+b})(1-\frac{b-c}{b+c})(1-\frac{c-a}{c+a})=\\\\=\frac{a+b-a+b}{a+b}\cdot \frac{b+c-b+c}{b+c}\cdot \frac{c+a-c+a}{c+a}=\frac{2b}{a+b}\cdot \frac{2c}{b+c}\cdot \frac{2a}c+a}=\frac{8\, abc}{(a+b)(b+c)(c+a)}\\\\\\\Rightarrow \; \; \; (1+x)(1+y)(1+z)=(1-x)(1-y)(1-z)$

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