446407Aſſumpt. Liber.
micirculis A C, B D, ergo ſi auferamus ex illis duos ſemicirculos A C,
B D, qui ſunt communes, remanet figura contenta à quatuor ſemicircu-
lis A B, C D, D B, A C, (quæ ea eſt, quàm vocat Archimedes Sali-
non) æqualis circulo, cuius diameter eſt F G, & hoc eſt quod voluimus.
B D, qui ſunt communes, remanet figura contenta à quatuor ſemicircu-
lis A B, C D, D B, A C, (quæ ea eſt, quàm vocat Archimedes Sali-
non) æqualis circulo, cuius diameter eſt F G, & hoc eſt quod voluimus.
PROPOSITIO XV.
SI fuerit A B ſemicirculus, &
A C corda Pentagoni, &
ſe-
miſſis arcus A C ſit A D, iungatur C D, & producatur
vt cadat ſuper E, & iungatur D B, quæ ſecet C A in F, &
ducatur ex F perpendicularis F G ſuper A B, erit linea E G
æqualis ſemidiametro circuli.
miſſis arcus A C ſit A D, iungatur C D, & producatur
vt cadat ſuper E, & iungatur D B, quæ ſecet C A in F, &
ducatur ex F perpendicularis F G ſuper A B, erit linea E G
æqualis ſemidiametro circuli.
Iungamus itaque lineam C B, &
ſit centrum H, &
iungamus H D,
D G, & A D. Et quia angulus A B C, cuius baſis eſt latus Pentagoni,
eſt duæ quintæ partes recti, quilibet duorum angulorum C B D, D B
A eſt quinta pars recti, & angulus D H A duplus eſt anguli D B H,
ergo angulus D H A eſt duæ quinte partes recti. Et quia in duobus trian-
gulis C B F, G B F duo anguli B ſunt æquales, & G, C recti, & latus
F B commune, erit B C æquale ipſi B G: & quia in duobus triangulis
C B D, G B D duo latera C B, B G ſunt æqualia, & ſimiliter duo an-
guli ad B, & latus B D commune, erunt duo anguli B C D, B G D
æquales, & quilibet eorum eſt ſex quintæ partes recti, & eſt æqualis an-
gulo D A E externo quadrilateri B A D C, quod eſt in circulo, ergo
remanet angulus D A B æqualis angulo D G A, & erit D A æqualis ip-
ſi D G. Et quia angulus D H G eſt duæ quintæ partes recti, & angulus
D G H ſex quintæ partes recti, remanet angulus H D G duæ quintæ par-
tes recti, & erit D G æqualis G H. Et quia A D E externus quadrila-
teri A D C B, quod eſt in circulo, eſt æqualis angulo C B A, &
D G, & A D. Et quia angulus A B C, cuius baſis eſt latus Pentagoni,
eſt duæ quintæ partes recti, quilibet duorum angulorum C B D, D B
A eſt quinta pars recti, & angulus D H A duplus eſt anguli D B H,
ergo angulus D H A eſt duæ quinte partes recti. Et quia in duobus trian-
gulis C B F, G B F duo anguli B ſunt æquales, & G, C recti, & latus
F B commune, erit B C æquale ipſi B G: & quia in duobus triangulis
C B D, G B D duo latera C B, B G ſunt æqualia, & ſimiliter duo an-
guli ad B, & latus B D commune, erunt duo anguli B C D, B G D
æquales, & quilibet eorum eſt ſex quintæ partes recti, & eſt æqualis an-
gulo D A E externo quadrilateri B A D C, quod eſt in circulo, ergo
remanet angulus D A B æqualis angulo D G A, & erit D A æqualis ip-
ſi D G. Et quia angulus D H G eſt duæ quintæ partes recti, & angulus
D G H ſex quintæ partes recti, remanet angulus H D G duæ quintæ par-
tes recti, & erit D G æqualis G H. Et quia A D E externus quadrila-
teri A D C B, quod eſt in circulo, eſt æqualis angulo C B A, &
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