Viviani, Vincenzo - De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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288 ſitque ipſum AB, quod ſecetur quacunque recta linea FN intra angulum.
BAC, efficient angulum AFN æquale angulo ACB. Iam dico rectam FN
productam cum reliquo latere AC conuenire, cumque baſi BC ad partem
minoris lateris AC.

Quoniam cum in triangulo BAC ſint anguli A, C, minores duobus rectis,
permutato C in F, erunt anguli FAC, AFN duobus rectis minores, ex quo
rectæ AC, FN conuenient ſimul in H, & reliquus angulus ABC in triangu-
lo BAC æquabitur reliquo angulo AHF in triangulo HAF, hoc eſt triangu-
la BAC, HAF erunt ſub contrariè poſita. Amplius cum ſit BA maior AC,
erit HA maior AF, propter ſimilitudinem triangulorum BAC, HAF, vnde
angulus AFH erit maior angulo AHF, ſiue angulo ABC, ſed anguli BFH,
AFH ſunt duobus rectis æquales, quare anguli BFH, ABC minores erunt
duobus rectis, ideoqne FH, BC ſimul conuenient, vt in G. Nunc verò con-

5[Figure 5] cipiatur per rectam FHG duci planum ſecans triangulum per axem ABC,
communiſque ſectio huius ſecantis plani cum plano baſis coni ſit recta DGE
perpendicularis baſi BC trianguli per axem, & cum conica ſuperficie ſectio-
nem efficiens MFTH, cuius diameter ſit FH. Itaque iam ſuperiùs oſtenſum
eſt, ſi fiat vt rectangulum FGH ad rectangulum BGC, ita diameter FH ad
aliam lineam FL, quæ ex F ordinatim in ſectione ductis æquidiſtet, iunga-
turque HL, quadratum cuiuſcunque applicatæ MN parallelæ communi ſe-
ctioni DE, æquari rectangulo NP, applicato rectę FL deficientique rectan-
gulo LX ſimili rectangulo ſub HF, FL. Quod verò talia latera HF, FL inter
ſe ſint æqualia ita oſtenditur.

Cum ſit enim angulus AFH æqualis angulo ACB, erit conſequens BFG
conſequenti HCG æqualis, eſtque angulus BGF æqualis angulo HGC, cum
in tertia figura idem ſint, in quarta verò ſint ad verticem, quare in triangu-
lis BGF; HGC circa æquales angulos ad G erunt latera proportionalia, ſiue
vt FG ad GB ita CG ad GH, ideoque rectangulum FGH æquale erit rectan-
gulo BGC, ſed vt rectangulum FGH ad BGC, ita tranſuerſum HF ad

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