postérieurement par M. Lhuilier. (Annales, tom. I, pag. 150)[1].
Pour le triangle sphérique, on aurait
![{\displaystyle \operatorname {Sin} .{\frac {1}{2}}T={\frac {\sqrt {\operatorname {Tang} .\alpha \operatorname {Tang} .\beta \operatorname {Tang} .\gamma \operatorname {Tang} .r}}{2\operatorname {Sin} .{\frac {1}{2}}a.\operatorname {Sin} .{\frac {1}{2}}b.\operatorname {Sin} .{\frac {1}{2}}c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91aa0cbf53e94d143af0baf8fe91d585227adefc)
Si de l’équation (3) on élimine tour à tour les quatre rayons, au moyen de la relation (2), on trouvera
![{\displaystyle T^{2}={\frac {\alpha ^{2}\beta ^{2}\gamma ^{2}}{\beta \gamma +\gamma \alpha +\alpha \beta }}=r^{2}.{\frac {\beta ^{2}\gamma ^{2}}{\beta \gamma -r(\beta +\gamma )}}=r^{2}.{\frac {\gamma ^{2}\alpha ^{2}}{\gamma \alpha -r(\gamma +\alpha )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62d11ef4a2aa662ce621dfec4df4494d128f2c40)
![{\displaystyle =r^{2}.{\frac {\alpha ^{2}\beta ^{2}}{\alpha \beta -r(\alpha +\beta )}}\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/180a9ae3a3908f3ba3847033cfe05adeda339a53)
(4)
Des équations (1) on tire (3)
![{\displaystyle \left.{\begin{aligned}&a={\frac {\alpha -r}{\alpha r}}.T=(\alpha -r){\sqrt {\frac {\beta \gamma }{\alpha r}}},\\\\&b={\frac {\beta -r}{\beta r}}.T=(\beta -r){\sqrt {\frac {\gamma \alpha }{\beta r}}},\\\\&c={\frac {\gamma -r}{\gamma r}}.T=(\gamma -r){\sqrt {\frac {\alpha \beta }{\gamma r}}}\,;\end{aligned}}\right\}\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0307a4274c037ab5fea56689e51b515a27a52155)
(5)
d’où
![{\displaystyle {\frac {a\alpha }{\alpha -r}}={\frac {b\beta }{\beta -r}}={\frac {c\gamma }{\gamma -r}}={\frac {T}{r}}\,;\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c47ccbdfd557984173e6746e3cfd360fadb684)
(6)
et par suite (3)
- ↑ Ce théorème fait aussi partie de la note de M. Bobillier.
J. D. G.