AS

ABC

\frac{bc\sqrt{2b^{2}+2c^{2}-a^{2}}}{b^{2}+c^{2}}

AB=c

AC=b

BC=a

AM

ABC

\angle BAS=\angle CAM

\frac{BS}{CM}=\frac{S_{\triangle BAS}}{S_{\triangle CAM}}=\frac{\frac{1}{2}AB\cdot AS\sin\angle BAS}{\frac{1}{2}AC\cdot AM\sin\angle CAM}=\frac{AB\cdot AS}{AC\cdot AM},

\frac{AS}{AM}=\frac{BS}{CM}\cdot\frac{AC}{AB}=\frac{BS\cdot b}{\frac{a}{2}\cdot c}=\frac{2b\cdot BS}{ac}.

\frac{BS}{CS}=\frac{c^{2}}{b^{2}}

\frac{BS}{BC}=\frac{c^{2}}{b^{2}+c^{2}}

BS=\frac{c^{2}BC}{b^{2}+c^{2}}=\frac{ac^{2}}{b^{2}+c^{2}},

2AM=\sqrt{2b^{2}+2c^{2}-a^{2}}

AS=\frac{2b\cdot BS\cdot AM}{ac}=\frac{b\cdot\frac{ac^{2}}{b^{2}+c^{2}}\cdot\sqrt{2b^{2}+2c^{2}-a^{2}}}{ac}=\frac{bc\sqrt{2b^{2}+2c^{2}-a^{2}}}{b^{2}+c^{2}}.

AB^{2}\cdot CS+AC^{2}\cdot BS-AS^{2}\cdot BC=BC\cdot BS\cdot CS,

\frac{BS}{CS}=\frac{AB^{2}}{AC^{2}}=\frac{c^{2}}{b^{2}}

BS=BC\cdot\frac{c^{2}}{b^{2}+c^{2}}=\frac{ac^{2}}{b^{2}+c^{2}},~CS=BC\cdot\frac{b^{2}}{b^{2}+c^{2}}=\frac{ab^{2}}{b^{2}+c^{2}},

c^{2}\cdot\frac{ab^{2}}{b^{2}+c^{2}}+b^{2}\cdot\frac{ac^{2}}{b^{2}+c^{2}}-AS^{2}\cdot a=a\cdot\frac{ac^{2}}{b^{2}+c^{2}}\cdot\frac{ab^{2}}{b^{2}+c^{2}},

AS^{2}=\frac{2b^{2}c^{2}}{b^{2}+c^{2}}-\frac{a^{2}b^{2}c^{2}}{(b^{2}+c^{2})^{2}}=\frac{b^{2}c^{2}(2b^{2}+2c^{2}-a^{2})}{(b^{2}+c^{2})^{2}}.

AS=\frac{bc\sqrt{2b^{2}+2c^{2}-a^{2}}}{b^{2}+c^{2}}.