COMMENTATIO ad QUAESTIONEM MATHLMATIGAM. 43 



_ ab[{a+ bY—c^2 _ ab{a + l)+c)(a+b --c) 

 {a + by {a+by " 



ergo : CD'" r= V("-f^ +c) (a + ^ - c>^ _ _2L x f/~ — "^W a bry 



a + b a-\-b f/^ ry l^a + bjry' 



CoBOLLARIUM. Inde sequitur esse: AE'" xBF"'xCD"=: 8i» : (/+m + « — r) 



Est cnim CD'''xBF'"x AE"' = 2lli"j2:x!Zj^£f:^X ^^^££? 



{a + b)ry {a + c)rß {b + c)r» 



__ ßPabcr\/r^.ßy Sl'^ab cr 



(a+6) {a+c) {b+c) r^aßy ~ (n+^.)(«+c)(6 + c)/-»/» 

 81'abc 

 = a6c(/ + ..+.-.) (Theor. XXXVII.) 



— 8^^ 



l+m-^-n — r 



P R O B L E M A II. Fig. n. 



Puncta F'" , D'" , E'" , rec/is si jungantur , oritur triangulum internum 



D"'F"'E"', cujus quaeruntur latera F"'D"' , D"'E"', E"'F'" , eorumaue se^menta 

 HF'" et HD'", ID'" et IE'", KE'" et KF'". 



SOLUTIO Part. I. Novimus ex elementis esse in triangulo AD"'F"': "*" '^~ 



D"'F"'^ = AD'"» +AF'"»— 2 AD'" xAF"'x COS. A - . ,. . 



{a+by ^ {a +cY (« +b) (a + c) ^ ''''*• ^ W 



necnonin iriang. ABC : BC» = AO + AB» — 2AC X AB X cos. A 



id est: a= = Ä> + c» — 26c X cos. A -i •;- j.;(j + \+ - 



quod si jnultiplicaris per --'^-___. ^ erit t^ — i^ -;- n; oü : aiigzailVils lijuuiaüo. 



Ka+b)[ci->rcy 



^°^c ^i ^c 5c3 25^e* 



(a + i) (a + c) (a + &) (a + c) + (a + 6)(a + c) " (a + 6) ,.{« ^ cj ^ '"*■ ^ ^^^ 

 quam aequationem si abstraxeris illi (a), reslabit : ''y^'lt~'" 

 xr _ _^!^_ =: ^'C' &»C^ & 3c 5c3 



(a + b){a+ c) (a + by + (a.+ c)» , (a + 6J ( a+ c) (a + 6)(a+c) 

 unde erit : \is+ 44. (^ _, i 4. ^^ (a 4- ,5 -|_n)] ^ ^^ ^ 



= 6»c» [(« + by + (a + c)^].— ic {a 4- 6) (a + c) (ä» + c^_ a»_) 

 adde: o =iV £2(fl+ö) (a-j-c)] _ 6c(a + 6)(rt+ c) (aöc) '" — "t "•- 

 efficitur : 



*»(a+Z.)»(a+c)'= b-c- [Ca + + (ß + c)]» - öc (a + ö) (« + c) [(04- cV— a»] 

 = b^c\2a -i-b + cy—bc {a + b){a + c){a+b+c)ib + c-a) 



F 2 et 



