{a^2-(b-c)^2}/{(a+c)^2-b^2} + {b^2-(a-c)^ 2}/{(a+b)^2 -c^2}+ {c^2-(a-b)^2}/{(b+c)^2 -a^2}
You asked:
MathBot Answer:
Evaluated
\(\displaystyle \frac{{a}^{2} - {\left( b - c \right)}^{2}}{{\left( a + c \right)}^{2} - {b}^{2}} + \frac{{b}^{2} - {\left( a - c \right)}^{2}}{{\left( a + b \right)}^{2} - {c}^{2}} + \frac{{c}^{2} - {\left( a - b \right)}^{2}}{{\left( b + c \right)}^{2} - {a}^{2}} = \frac{a^{2} - \left(b - c\right)^{2}}{- b^{2} + \left(a + c\right)^{2}} + \frac{b^{2} - \left(a - c\right)^{2}}{- c^{2} + \left(a + b\right)^{2}} + \frac{c^{2} - \left(a - b\right)^{2}}{- a^{2} + \left(b + c\right)^{2}} \)
Expanded
\[\frac{{a}^{2} - {\left( b - c \right)}^{2}}{{\left( a + c \right)}^{2} - {b}^{2}} + \frac{{b}^{2} - {\left( a - c \right)}^{2}}{{\left( a + b \right)}^{2} - {c}^{2}} + \frac{{c}^{2} - {\left( a - b \right)}^{2}}{{\left( b + c \right)}^{2} - {a}^{2}} = \frac{a^{2}}{a^{2} + 2 a c - b^{2} + c^{2}} - \frac{a^{2}}{a^{2} + 2 a b + b^{2} - c^{2}} - \frac{a^{2}}{- a^{2} + b^{2} + 2 b c + c^{2}} + \frac{2 a b}{- a^{2} + b^{2} + 2 b c + c^{2}} + \frac{2 a c}{a^{2} + 2 a b + b^{2} - c^{2}} - \frac{b^{2}}{a^{2} + 2 a c - b^{2} + c^{2}} + \frac{b^{2}}{a^{2} + 2 a b + b^{2} - c^{2}} - \frac{b^{2}}{- a^{2} + b^{2} + 2 b c + c^{2}} + \frac{2 b c}{a^{2} + 2 a c - b^{2} + c^{2}} - \frac{c^{2}}{a^{2} + 2 a c - b^{2} + c^{2}} - \frac{c^{2}}{a^{2} + 2 a b + b^{2} - c^{2}} + \frac{c^{2}}{- a^{2} + b^{2} + 2 b c + c^{2}}\]
Factored
\[\frac{{a}^{2} - {\left( b - c \right)}^{2}}{{\left( a + c \right)}^{2} - {b}^{2}} + \frac{{b}^{2} - {\left( a - c \right)}^{2}}{{\left( a + b \right)}^{2} - {c}^{2}} + \frac{{c}^{2} - {\left( a - b \right)}^{2}}{{\left( b + c \right)}^{2} - {a}^{2}} = 1\]