COMMENTATIO ad QUAESTIOIs'EM MATHEMATICAINL 



iguur (a + c — b) {a + b —c) + (b + c — a) (a + b — c) + (b + c—a) {a + c — b) = 



2ab + 2ac ■+■ 2bc — a^ — 6' — c". 



cjuod si per (a -\- b •{- c) mulliplicaveris , habebis : 



(^a+b+c)(^2ab+2ac+2bc — a= — £> — c'-)-=.&abc+a'^b+ab^-\-a^c+ac'+b'^c+bc^—a'—b'^—c^ 



. . „ Gabc + a^b + ab''+a''c+ac"- + b'-c + bc' — a^ — b^ — c^ ... 



indaque : oi + ß + y=. ^j . • • (.A^J 



Jam esse: r = , constat ex Theor. II, quod si duxeris in : 



a -\- b -\- c 



(b + c — n){a + c — b){ a +b — c) . _2 lib + c — a){a + c — b){a + b — c) 



(6 4-c — aj(a + c — 6)(a + Ä— c)' *^'** ''" 16/' 



{ b + c — a){a + c— b)(a + b — c) 



~ 8/ 

 ' , . _ — 2aJc+a>£4-ai>+a=c+ac»+i=c+6c*— a3— 63 — c3 

 •(jiiod reductione lit : /•=: " ' «/- ' ' — ' 



cui si addideris ; 4Ä = -j- = -kj- (Theor. I Coroll,) 



, .„ 6abc-{-a'b+ab^+a'c + ac^+b-c + bc' — a^ — J3— c* 

 invenies: 4Ä + r = 5-j: • • (l>) 



Ol 



eandem expressionem , quam reperimus ( A ) , ex quibus inter se collatis concludas 

 licet, esse : 



Co Ro LI, A RiuM. Inde colligi potest : summam radiorum circulorum exin- 

 icriptofuni ductnm in radium quadruplicem circuli inscripti , esse aequalem re- 

 siduo , quad reünquitur , duplici summae Jactorum binorum trianguli laterum 

 si demseris summam laterum quadratorum. Id est: 4r(i» + /3 + y) = 

 2 {ab + ac + bc) — (o* + 6=* + c=). 



In demonstrationis enim tbeorematis antecedentis progessu vidimus esse ; 



«+/3+y = "-±1^15 X i^iflb + ac + bc)- (a= + i« + c=)] 



Ol 



= 47 X [ 2 (a5 + «c + 6c) — («» + i'' + C) ] (Theor. II.) 

 ergo est: 4r (« + /3 + y)=: 2 (a5 + ac + 6c) — (a> + ä» + c») 



- THEOREMA V. Fig. 3. 



Trians^uli ABC latus quodlibet ad circuli inscripti radium ita refertur , ut Co- 

 sinus dimidii angull , qui est lateri Uli oppositus , ad factum sinuum dimidii re— 

 iiguorum angulorum. Ut sit v. ci AB : r = cos. \C : sin. |A . sin. |B. 



Demonstratio, Constat ex elementis trigonometricis , trianguli cuj uslibet la- 



-3HT 



