\(\sqrt { B^{ 2 }+r^{ 2 } } +\sqrt { C^{ 2 }+r^{ 2 } } -C-B=A\)
\(\sqrt { B^{ 2 }+r^{ 2 } } +\sqrt { C^{ 2 }+r^{ 2 } } =A+B+C\)
\( \sqrt { B^{ 2 }+r^{ 2 } } -\sqrt { C^{ 2 }+r^{ 2 } } =A+B+C-2\sqrt { C^{ 2 }+r^{ 2 } } \)
\( B^{ 2 }-C^{ 2 }=(A+B+C)(A+B+C-2\sqrt { C^{ 2 }+r^{ 2 } } )\)
\( B^{ 2 }-C^{ 2 }=(A+B+C)^{ 2 }-2(A+B+C)\sqrt { C^{ 2 }+r^{ 2 } } \)
\( \sqrt { C^{ 2 }+r^{ 2 } } =\cfrac { (A+B+C)^{ 2 }-B^{ 2 }+C^{ 2 } }{ 2(A+B+C) } \)
\( r^{ 2 }=\cfrac { ((A+B+C)^{ 2 }-B^{ 2 }+C^{ 2 })^{ 2 }-4C^{ 2 }(A+B+C)^{ 2 } }{ 4(A+B+C)^{ 2 } } \)
\( =\cfrac { \left\{ (A+B+C)^{ 2 }-B^{ 2 }+C^{ 2 }+2C(A+B+C) \right\}. \\ \quad \left\{ (A+B+C)^{ 2 }-B^{ 2 }+C^{ 2 }-2C(A+B+C) \right\} }{ 4(A+B+C)^{ 2 } } \)
\( =\cfrac { \left\{ (A+C)^{ 2 }+2B(A+C)+C^{ 2 }+2C(A+B+C) \right\}.\\ \quad \left\{ (A+C)^{ 2 }+2B(A+C)+C^{ 2 }-2C(A+B+C) \right\} }{ 4(A+B+C)^{ 2 } } \)
\( r^{ 2 }=\cfrac { \left\{ A(A+2C)+2C(A+2C)+2B(A+2C) \right\}A\left\{ A+2B \right\} }{ 4(A+B+C)^{ 2 } } \)
\( r^{ 2 }=\cfrac { A\left\{ A+2B \right\} \left\{ A+2C \right\} \left\{ A+2(B+C) \right\} }{ 4(A+B+C)^{ 2 } } \)
\( r=\cfrac { \sqrt { A\left\{ A+2B \right\} \left\{ A+2C \right\} \left\{ A+2(B+C) \right\} } }{ 2(A+B+C) } \)
\( r\approx\cfrac { \sqrt { A(2B)(2C)(2(B+C)) } }{ 2(B+C) } \)
\( r\approx\sqrt { \cfrac { { 2ABC } }{ B+C } } ,\quad \quad A=\cfrac { \lambda }{ 2 } \)
\( r\approx\sqrt { \lambda \cfrac { { BC } }{ B+C } } \)
The carrier destructively interfered. Why would there be alternate paths, other than the straight line?
