Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/376

$aa'\operatorname {Cos} .(A-X)+bb'\operatorname {Cos} .(B-X)+cc'\operatorname {Cos} .(C-X)=0,$

ou bien en développant

$(aa'\operatorname {Cos} .A+bb'\operatorname {Cos} .B+cc'\operatorname {Cos} .C)\operatorname {Cos} .X$

$+(aa'\operatorname {Sin} .A+bb'\operatorname {Sin} .B+cc'\operatorname {Sin} .C)\operatorname {Sin} .X=0\,;$

d’où

$\operatorname {Tang} .X=-{\frac {aa'\operatorname {Cos} .A+bb'\operatorname {Cos} .B+cc'\operatorname {Cos} .C}{aa'\operatorname {Sin} .A+bb'\operatorname {Sin} .B+cc'\operatorname {Sin} .C}}.\qquad$(14)

La première des équations (12) donne

$\lambda ={\frac {p-2a}{2a\operatorname {Cos} .(A-X)}}\,;$

mais de la valeur de $\operatorname {Tang} .X$ on conclut aisément

$\operatorname {Sin} .X=$

$-{\frac {aa'\operatorname {Cos} .A+bb'\operatorname {Cos} .B+cc'\operatorname {Cos} .C}{\sqrt {a^{2}a'^{2}+b^{2}b'^{2}+c^{2}c'^{2}+2bb'cc'\operatorname {Cos} .A'+2cc'aa'\operatorname {Cos} .B'+2aa'bb'\operatorname {Cos} .C'}}},$

$\operatorname {Cos} .X=$

$-{\frac {aa'\operatorname {Sin} .A+bb'\operatorname {Sin} .B+cc'\operatorname {Sin} .C}{\sqrt {a^{2}a'^{2}+b^{2}b'^{2}+c^{2}c'^{2}+2bb'cc'\operatorname {Cos} .A'+2cc'aa'\operatorname {Cos} .B'+2aa'bb'\operatorname {Cos} .C'}}},$

et par suite

$\operatorname {Cos} .A\operatorname {Cos} .X+\operatorname {Sin} .A\operatorname {Sin} .X=\operatorname {Cos} .(A-X)$

$={\frac {cc'\operatorname {Sin} .B'-bb'\operatorname {Sin} .C'}{\sqrt {a^{2}a'^{2}+b^{2}b'^{2}+c^{2}c'^{2}+2bb'cc'\operatorname {Cos} .A'+2cc'aa'\operatorname {Cos} .B'+2aa'bb'\operatorname {Cos} .C'}}}.$

D’un autre côté la formule (13) donne

$p-2a=-2a.{\frac {cc'\operatorname {Sin} .B'-bb'\operatorname {Sin} .C'}{bc\operatorname {Sin} .A'+ca\operatorname {Sin} .B'+ab\operatorname {Sin} .C'}}\,;$