338152
LEMMA IV. PROP. VIII.
Si in triangulo A B C, cuius baſis A B, ex vertice C ducta ſit
C E ipſi B A parallela, vel ad eaſdem, vel ad oppoſitas partes, &
ducatur quælibet A D E vtranque B C, C E ſecans in D, & E:
dico aggregatum triangulorum A D B, D C E ad triangulum A
C B eſſe vt aggregatum extremarum poſt B D, D C, ad B C.
C E ipſi B A parallela, vel ad eaſdem, vel ad oppoſitas partes, &
ducatur quælibet A D E vtranque B C, C E ſecans in D, & E:
dico aggregatum triangulorum A D B, D C E ad triangulum A
C B eſſe vt aggregatum extremarum poſt B D, D C, ad B C.
SVmatur D F tertia proportionalis poſt B D, D C.
Iam triangulum D C E ad A D C eſt vt E D ad D A, vel vt C D ad
269[Figure 269]
D B, vel vt D F ad D C;
&
triangulum A D C ad trian-
gulum A B C, eſt vt D C
ad C B, ergo ex æquali triã-
gulum D C E ad A B C, erit
vt D F ad C B; ſed triangu-
lum A D B ad idem A B C
eſt vt B D ad B C, quare
duo ſimul triangula D C E,
A D B, ad triangulum A C
B, erunt vt duæ ſimul lineæ
D F, D B, hoc eſt tota B F,
aggregatum extremarum poſt B D, D C, ad B C. Quod erat, & c.
triangulum A D C ad trian-
gulum A B C, eſt vt D C
ad C B, ergo ex æquali triã-
gulum D C E ad A B C, erit
vt D F ad C B; ſed triangu-
lum A D B ad idem A B C
eſt vt B D ad B C, quare
duo ſimul triangula D C E,
A D B, ad triangulum A C
B, erunt vt duæ ſimul lineæ
D F, D B, hoc eſt tota B F,
aggregatum extremarum poſt B D, D C, ad B C. Quod erat, & c.
PROBL. III. PROP. IX.
Duabus datis rectis lineis terminatis ad quemlibet angulum
conſtitutis, & per vnius ipſarum terminum alia alteri datarum
æquidiſtanter ducta, ad contrarias tamen partes, & in infinitum
producta: oportet per extremum terminum alterius, rectam duce-
re ęquidiſtanti occurrentem, ita vt, cum ipſa bina ſimilia triangula
ad verticem conſtituat, horũ aggregatum ſit MINIMA quantitas.
conſtitutis, & per vnius ipſarum terminum alia alteri datarum
æquidiſtanter ducta, ad contrarias tamen partes, & in infinitum
producta: oportet per extremum terminum alterius, rectam duce-
re ęquidiſtanti occurrentem, ita vt, cum ipſa bina ſimilia triangula
ad verticem conſtituat, horũ aggregatum ſit MINIMA quantitas.
SInt A B, B C rectæ lineæ terminatæ ad quemcunque angulum A B C
compoſitæ, ſitque C D in infinitum producta ipſi B A parallela, ſed ad
oppoſit as partes rectæ C B: oportet ex A rectam ducere, qualis eſt A D,
ita vt aggregatum ſimilium triangulorum A E B, C E D ad verticem E
ſit _MINIMVM_.
compoſitæ, ſitque C D in infinitum producta ipſi B A parallela, ſed ad
oppoſit as partes rectæ C B: oportet ex A rectam ducere, qualis eſt A D,
ita vt aggregatum ſimilium triangulorum A E B, C E D ad verticem E
ſit _MINIMVM_.
Diuidatur B C in E, ita vt B E ad E C ſit vt latus cuiuſdam quadrati ad
exceſſum diametri ſuper latus: dico punctum E eſſe quæſitum.
exceſſum diametri ſuper latus: dico punctum E eſſe quæſitum.
Nam ducta qualibet alia A F G;
iunctaque A C:
cum aggregatum ex-
tremarum proportionalium poſt B E, E C ſit _MINIMVM_ (per
tremarum proportionalium poſt B E, E C ſit _MINIMVM_ (per