444405Aſſump. Liber.
immo duo anguli A B F, A C F minores ſint duobus angulis A B D,
A C D, totum ſua parte, & hoc eſt abſurdum, ergo manet propoſitum.
A C D, totum ſua parte, & hoc eſt abſurdum, ergo manet propoſitum.
Notæ in Propoſit. XII.
LEmma aſſumptum in demonſtratione huius pulcherrimæ propoſitionis poteſt
directè oſtendi hac ratione.
directè oſtendi hac ratione.
Si in quadrilatero A C D B duo latera A C, &
A B æqualia fuerint, atque
angulus C D B æqualis duobus angulis C, & B ſimul ſumptis. Dico rectam A
D ipſi A C, vel A B æqualẽ eſſe. Producatur C A, in E, vt A E fiat æqualis
A B, iungaturque B E. Quia in triangulo Iſo-
518[Figure 518]
ſcelio B A E angulus E æqualis eſt angulo A B
E, & angulus C D B æqualis eſt duobus angulis
C, & D B A ſimul ſumptis, ergo duo anguli C D
B, & E (oppoſiti in quadrilatero C D B E)
æquales ſunt tribus angulis C, D B A, & A B
E, ſeu duobus angulis C, & D B E, ſed qua-
tuor anguli quadrilateri E C D B æquales ſunt
quatuor rectis, ergo duo anguli oppoſiti E, C D
B duobus rectis æquales ſunt, & propterea qua-
drilaterum ipſum circulo inſcribi poteſt, cuius
circuli centrum erit A, cum tres rectæ lineæ
C A, A B, A E æquales poſitæ ſint, & propte-
rea A D radius quoque circuli erit æqualis ipſi C A.
angulus C D B æqualis duobus angulis C, & B ſimul ſumptis. Dico rectam A
D ipſi A C, vel A B æqualẽ eſſe. Producatur C A, in E, vt A E fiat æqualis
A B, iungaturque B E. Quia in triangulo Iſo-
E, & angulus C D B æqualis eſt duobus angulis
C, & D B A ſimul ſumptis, ergo duo anguli C D
B, & E (oppoſiti in quadrilatero C D B E)
æquales ſunt tribus angulis C, D B A, & A B
E, ſeu duobus angulis C, & D B E, ſed qua-
tuor anguli quadrilateri E C D B æquales ſunt
quatuor rectis, ergo duo anguli oppoſiti E, C D
B duobus rectis æquales ſunt, & propterea qua-
drilaterum ipſum circulo inſcribi poteſt, cuius
circuli centrum erit A, cum tres rectæ lineæ
C A, A B, A E æquales poſitæ ſint, & propte-
rea A D radius quoque circuli erit æqualis ipſi C A.
PROPOSITIO XIII.
SI mutuo ſe ſecent duæ lineæ A B, C D in circulo, &
fue-
rit A B diameter illius, at non C D, & educantur ex duo-
bus punctis A, B duæ per-
519[Figure 519]
pendiculares ad C D, quæ
ſint A E, B F, vtique ab-
ſcindent ex illa C F, D E
æquales.
rit A B diameter illius, at non C D, & educantur ex duo-
bus punctis A, B duæ per-
ſint A E, B F, vtique ab-
ſcindent ex illa C F, D E
æquales.
Iungamus E B, &
educamus
ex I, quod eſt centrum, per-
pendicularem I G ſuper C D,
& producamus eam ad H in E
B. Et quia I G eſt perpendicu-
laris ex centro ad C D illam bi-
fariam diuidet in G, & quia I
G, A E ſunt duæ perpendicu-
lares ſuper illam, erunt
ex I, quod eſt centrum, per-
pendicularem I G ſuper C D,
& producamus eam ad H in E
B. Et quia I G eſt perpendicu-
laris ex centro ad C D illam bi-
fariam diuidet in G, & quia I
G, A E ſunt duæ perpendicu-
lares ſuper illam, erunt