Teorema di Nepero
# Teorema di Nepero

In matematica, il teorema di Nepero afferma le seguenti identità, utilizzando la notazione standard per gli elementi di un triangolo:

    ${\frac {b+c}{b-c}}={\frac {\tan {\displaystyle {\frac {\beta +\gamma }{2}}}}{\tan {\displaystyle {\frac {\beta -\gamma }{2}}}}}$
    ${\frac {c+a}{c-a}}={\frac {\tan {\displaystyle {\frac {\gamma +\alpha }{2}}}}{\tan {\displaystyle {\frac {\gamma -\alpha }{2}}}}}$
    ${\frac {a+b}{a-b}}={\frac {\tan {\displaystyle {\frac {\alpha +\beta }{2}}}}{\tan {\displaystyle {\frac {\alpha -\beta }{2}}}}}$

## Dimostrazione

Siano $a$, $b$, $c$ i lati di un triangolo, e siano $\alpha$, $\beta$, $\gamma$ gli angoli opposti, rispettivamente.

${\frac {b+c}{b-c}}={\frac {(b+c)^{2}}{(b+c)(b-c)}}={\frac {a^{2}(b+c)^{2}}{a^{2}(b+c)(b-c)}}={\frac {a^{2}}{b^{2}-c^{2}}}\left({\frac {b+c}{a}}\right)^{2}={\frac {(a^{2}+b^{2}-c^{2})+(a^{2}-b^{2}+c^{2})}{(a^{2}+b^{2}-c^{2})-(a^{2}-b^{2}+c^{2})}}\left({\frac {b+c}{a}}\right)^{2}$

Per il teorema dei seni $a=2R\sin \alpha$, $b=2R\sin \beta$ e $c=2R\sin \gamma$. Sostituendo si ottiene:

${\frac {(a^{2}+b^{2}-c^{2})+(a^{2}-b^{2}+c^{2})}{(a^{2}+b^{2}-c^{2})-(a^{2}-b^{2}+c^{2})}}\left({\frac {b+c}{a}}\right)^{2}={\frac {(\sin ^{2}\alpha +\sin ^{2}\beta -\sin ^{2}\gamma )+(\sin ^{2}\alpha -\sin ^{2}\beta +\sin ^{2}\gamma )}{(\sin ^{2}\alpha +\sin ^{2}\beta -\sin ^{2}\gamma )-(\sin ^{2}\alpha -\sin ^{2}\beta +\sin ^{2}\gamma )}}\left({\frac {\sin \beta +\sin \gamma }{\sin \alpha }}\right)^{2}$ (1)

Consideriamo il secondo membro: usando le formule di prostaferesi, la formula di duplicazione del seno e l'identità $\displaystyle \alpha +\beta +\gamma =\pi$ diventa

$\left({\frac {\displaystyle 2\sin {\frac {\beta +\gamma }{2}}\cos {\frac {\beta -\gamma }{2}}}{\displaystyle 2\sin {\frac {\beta +\gamma }{2}}\cos {\frac {\beta +\gamma }{2}}}}\right)^{2}=\left({\frac {\displaystyle \cos {\frac {\beta -\gamma }{2}}}{\displaystyle \cos {\frac {\beta +\gamma }{2}}}}\right)^{2}$

Consideriamo il primo addendo del numeratore del primo membro: $\displaystyle \sin ^{2}\alpha +\sin ^{2}\beta -\sin ^{2}\gamma$

Usando la formula di bisezione del coseno, le formule di prostaferesi, le identità $\displaystyle \alpha +\beta +\gamma =\pi$ e $\displaystyle \cos(\pi /2-x)=\sin x$, otteniamo:

$\sin ^{2}\alpha +\sin ^{2}\beta -\sin ^{2}\gamma =1-\cos ^{2}\alpha +1-\cos ^{2}\beta -1+\cos ^{2}\gamma =1-{\frac {\cos 2\alpha +1}{2}}-{\frac {\cos 2\beta +1}{2}}+\cos ^{2}\gamma =\cos ^{2}\gamma -{\frac {1}{2}}(\cos 2\alpha +\cos 2\beta )=$ $=\cos ^{2}\gamma +\cos \gamma \cos(\alpha -\beta )=\cos \gamma (\cos \gamma +\cos(\alpha -\beta ))=2\cos \gamma \cos {\frac {\alpha -\beta +\gamma }{2}}\cos {\frac {-\alpha +\beta +\gamma }{2}}=2\sin \alpha \sin \beta \cos \gamma$

Allo stesso modo si ottiene che

$\displaystyle \sin ^{2}\alpha -\sin ^{2}\beta +\sin ^{2}\gamma =2\sin \alpha \cos \beta \sin \gamma$

Sostituendo le espressioni trovate per il primo e il secondo membro nella (1) e usando la formula di somma del seno, otteniamo

${\frac {b+c}{b-c}}={\frac {2\sin \alpha \sin \beta \cos \gamma +2\sin \alpha \cos \beta \sin \gamma }{2\sin \alpha \sin \beta \cos \gamma -2\sin \alpha \cos \beta \sin \gamma }}\left({\frac {\displaystyle \cos {\frac {\beta -\gamma }{2}}}{\displaystyle \cos {\frac {\beta +\gamma }{2}}}}\right)^{2}={\frac {\sin(\beta +\gamma )}{\sin(\beta -\gamma )}}\left({\frac {\displaystyle \cos {\frac {\beta -\gamma }{2}}}{\displaystyle \cos {\frac {\beta +\gamma }{2}}}}\right)^{2}$

Usando la formula di duplicazione del seno otteniamo

${\frac {b+c}{b-c}}={\frac {\sin(\beta +\gamma )}{\sin(\beta -\gamma )}}\left({\frac {\displaystyle \cos {\frac {\beta -\gamma }{2}}}{\displaystyle \cos {\frac {\beta +\gamma }{2}}}}\right)^{2}={\frac {\displaystyle 2\sin {\frac {\beta +\gamma }{2}}\cos {\frac {\beta +\gamma }{2}}}{\displaystyle 2\sin {\frac {\beta -\gamma }{2}}\cos {\frac {\beta -\gamma }{2}}}}\left({\frac {\displaystyle \cos {\frac {\beta -\gamma }{2}}}{\displaystyle \cos {\frac {\beta +\gamma }{2}}}}\right)^{2}={\frac {\displaystyle \tan {\frac {\beta +\gamma }{2}}}{\displaystyle \tan {\frac {\beta -\gamma }{2}}}}$

