43 CAROLI JOHANNIS MATTHES, henr. fil. 



uiule : [a -j- 6) (a -f- c) (6 + c) : abc = l •{• m -^ n — r : r. 

 COROLLARIUM. Inde sequitur esse : (a +0} (« + cj (ö + c)= . 



THEOREMA XXXVIII. Fig. n. 



Partes AD'" ef BD'", BE"' c< CE'", CF'" et AF'" , //z (/««.s clividantur trian- 

 gulL ABC lalera AB,BC kC per rectas , qaae angidos dimidiant oppositos , 

 singulae aequales sunt qaolo facti laleris porlinni adjacenlis et ipslus , cujus 

 pars est, difiso per reliqnorum duorurii laterurn saiiunam, Ut alt: AD'" = 



i:£^,BD'"= -^,eic. 

 r? + 6 ' a + b 



Demonstratio. Quia dimidiat recta CD"' atigulum C, habemus ex nota 



tiianguli proprietate ( • ") : AD'" : BD'" = AC : BC = 6 : a 



unde compositione : AD'" : AD"' + BD' ' ■=: b : a + b> 



idest: AD'" : c = 6: a + i, unde : AD'" = — —, • 



a + b' 



item: BD'" -. KDV + BD'." = a: a + b , et hinc : BD"'= -21-. 



a + b 



GoROLtARiUM : AD'"x BE'" xCF"'= BD'" x CE'" X AF'" =4/iZr: {l+m+n — r) 

 Nam est ;, AD'" X BE!" X GF'" = -^ x -^ X "^ ""'^''^ 



a+b b + c a+c (a + b) {a + c) {b + c) 



l-\-m-\- n — r 



THEOREMA XXXIX. Fig. u. 



Rectae CD'", BF"', AE"', quae trianguli ABC dimidiant angulos , atque ab 

 altera parte terminantur lateribus subjectis , hunccä Jiabent calorem : 



CD"' = ^^^^^^^ BF'" _ sJ^V^c-ß ^^,„ _ iiybcra. 

 (a + b)ry* (a + c)rß ' {b + c)ra 



Demonstratio. Es parle altera ejusdem trianguli proprietate, qua usi su-^ 

 mus iu Theor. antecedenti, babemus in Iriangulo ABC :: 

 CD"'» _ AC X BC — AD'" X BD'" 



idest: QD"'- — ab-^^>i-^z=ab ^^^ (Theor. XXXVIII.) 



a+b a-^b {a + by ^ ' 



(,i.) CC J de Gelder, Beglns. der Meetk. Llb. IV. THeor. XIX. 



