Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{(a-b+c)(a-b-c)}{{a}^{2}-{(b+c)}^{2}}+\frac{{(b-c)}^{2}-{a}^{2}}{{b}^{2}-{(c+a)}^{2}}+\frac{{(c-a)}^{2}-{b}^{2}}{{c}^{2}-{(a+b)}^{2}}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{(a-b+c)(a-b-c)}{(a+b+c)(a-(b+c))}+\frac{{(b-c)}^{2}-{a}^{2}}{{b}^{2}-{(c+a)}^{2}}+\frac{{(c-a)}^{2}-{b}^{2}}{{c}^{2}-{(a+b)}^{2}}\]
Remove parentheses.
\[\frac{(a-b+c)(a-b-c)}{(a+b+c)(a-b-c)}+\frac{{(b-c)}^{2}-{a}^{2}}{{b}^{2}-{(c+a)}^{2}}+\frac{{(c-a)}^{2}-{b}^{2}}{{c}^{2}-{(a+b)}^{2}}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{(a-b+c)(a-b-c)}{(a+b+c)(a-b-c)}+\frac{(b-c+a)(b-c-a)}{{b}^{2}-{(c+a)}^{2}}+\frac{{(c-a)}^{2}-{b}^{2}}{{c}^{2}-{(a+b)}^{2}}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{(a-b+c)(a-b-c)}{(a+b+c)(a-b-c)}+\frac{(b-c+a)(b-c-a)}{(b+c+a)(b-(c+a))}+\frac{{(c-a)}^{2}-{b}^{2}}{{c}^{2}-{(a+b)}^{2}}\]
Remove parentheses.
\[\frac{(a-b+c)(a-b-c)}{(a+b+c)(a-b-c)}+\frac{(b-c+a)(b-c-a)}{(b+c+a)(b-c-a)}+\frac{{(c-a)}^{2}-{b}^{2}}{{c}^{2}-{(a+b)}^{2}}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{(a-b+c)(a-b-c)}{(a+b+c)(a-b-c)}+\frac{(b-c+a)(b-c-a)}{(b+c+a)(b-c-a)}+\frac{(c-a+b)(c-a-b)}{{c}^{2}-{(a+b)}^{2}}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{(a-b+c)(a-b-c)}{(a+b+c)(a-b-c)}+\frac{(b-c+a)(b-c-a)}{(b+c+a)(b-c-a)}+\frac{(c-a+b)(c-a-b)}{(c+a+b)(c-(a+b))}\]
Remove parentheses.
\[\frac{(a-b+c)(a-b-c)}{(a+b+c)(a-b-c)}+\frac{(b-c+a)(b-c-a)}{(b+c+a)(b-c-a)}+\frac{(c-a+b)(c-a-b)}{(c+a+b)(c-a-b)}\]
Cancel \(a-b-c\).
\[\frac{a-b+c}{a+b+c}+\frac{(b-c+a)(b-c-a)}{(b+c+a)(b-c-a)}+\frac{(c-a+b)(c-a-b)}{(c+a+b)(c-a-b)}\]
Cancel \(b-c-a\).
\[\frac{a-b+c}{a+b+c}+\frac{b-c+a}{b+c+a}+\frac{(c-a+b)(c-a-b)}{(c+a+b)(c-a-b)}\]
Cancel \(c-a-b\).
\[\frac{a-b+c}{a+b+c}+\frac{b-c+a}{b+c+a}+\frac{c-a+b}{c+a+b}\]
Simplify.
\[\frac{a-b+c}{a+b+c}+\frac{b-c+a}{a+b+c}+\frac{c-a+b}{a+b+c}\]
Join the denominators.
\[\frac{a-b+c+b-c+a+c-a+b}{a+b+c}\]
Collect like terms.
\[\frac{(a+a-a)+(-b+b+b)+(c-c+c)}{a+b+c}\]
Simplify \((a+a-a)+(-b+b+b)+(c-c+c)\) to \(a+b+c\).
\[\frac{a+b+c}{a+b+c}\]
Cancel \(a+b+c\).
\[1\]
1
