Solve frac{p^{2}-(q-r)^{2}}{(p+r)^{2}-q^{2}}+frac{q^{2}-(p-r)^{2}}{(p+q)^{2}-r^{2}}+r^2-(p-q)^2/(q+r)^2-p^2

$$\frac { p ^ { 2 } - ( q - r ) ^ { 2 } } { ( p + r ) ^ { 2 } - q ^ { 2 } } + \frac { q ^ { 2 } - ( p - r ) ^ { 2 } } { ( p + q ) ^ { 2 } - r ^ { 2 } } + \frac { r ^ { 2 } - ( p - q ) ^ { 2 } } { ( q + r ) ^ { 2 } - p ^ { 2 } }$$

Evaluate

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Short Solution Steps

Factor the expressions that are not already factored in \frac{p^{2}-\left(q-r\right)^{2}}{\left(p+r\right)^{2}-q^{2}}.

$$\frac{\left(p+q-r\right)\left(p-q+r\right)}{\left(p+q+r\right)\left(p-q+r\right)}+\frac{q^{2}-\left(p-r\right)^{2}}{\left(p+q\right)^{2}-r^{2}}+\frac{r^{2}-\left(p-q\right)^{2}}{\left(q+r\right)^{2}-p^{2}}$$

Cancel out p-q+r in both numerator and denominator.

$$\frac{p+q-r}{p+q+r}+\frac{q^{2}-\left(p-r\right)^{2}}{\left(p+q\right)^{2}-r^{2}}+\frac{r^{2}-\left(p-q\right)^{2}}{\left(q+r\right)^{2}-p^{2}}$$

Factor the expressions that are not already factored in \frac{q^{2}-\left(p-r\right)^{2}}{\left(p+q\right)^{2}-r^{2}}.

$$\frac{p+q-r}{p+q+r}+\frac{\left(p+q-r\right)\left(-p+q+r\right)}{\left(p+q+r\right)\left(p+q-r\right)}+\frac{r^{2}-\left(p-q\right)^{2}}{\left(q+r\right)^{2}-p^{2}}$$

Cancel out p+q-r in both numerator and denominator.

$$\frac{p+q-r}{p+q+r}+\frac{-p+q+r}{p+q+r}+\frac{r^{2}-\left(p-q\right)^{2}}{\left(q+r\right)^{2}-p^{2}}$$

Factor the expressions that are not already factored in \frac{r^{2}-\left(p-q\right)^{2}}{\left(q+r\right)^{2}-p^{2}}.

$$\frac{p+q-r}{p+q+r}+\frac{-p+q+r}{p+q+r}+\frac{\left(p-q+r\right)\left(-p+q+r\right)}{\left(p+q+r\right)\left(-p+q+r\right)}$$

Cancel out -p+q+r in both numerator and denominator.

$$\frac{p+q-r}{p+q+r}+\frac{-p+q+r}{p+q+r}+\frac{p-q+r}{p+q+r}$$

Since \frac{p+q-r}{p+q+r} and \frac{-p+q+r}{p+q+r} have the same denominator, add them by adding their numerators.

$$\frac{p+q-r-p+q+r}{p+q+r}+\frac{p-q+r}{p+q+r}$$

Combine like terms in p+q-r-p+q+r.

$$\frac{2q}{p+q+r}+\frac{p-q+r}{p+q+r}$$

Since \frac{2q}{p+q+r} and \frac{p-q+r}{p+q+r} have the same denominator, add them by adding their numerators.

$$\frac{2q+p-q+r}{p+q+r}$$

Combine like terms in 2q+p-q+r.

$$\frac{q+p+r}{p+q+r}$$

Cancel out p+q+r in both numerator and denominator.

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Factor

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