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

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140116

LEMMA IX. PROP. LXXII.

Dato angulo rectilineo ABC, cuius diameter BDE, & applica-
ta ADC: oportet ex C, infra ADC, ſecantem ducere CEF, ita vt
ſi ex E, & F applicentur EH, FG ipſi ADC parallelæ, quadratum
HE ad rectangulum DEG, datam habeaat rationem O ad P.

SVmatur Q media proportionalis inter O, & P; & ex D, niſi DA ſit dia-
metro BD perpendicularis, erigatur DL, & fiat vt O ad Q, ita DA ad
DL, iunctaque BLM, ſumatur LM æqualis LD, & per M demittatur ME per-
pendicularis ipſi BDE: dico per punctum E quæſitam ſecantem tranſire.

Nam iuncta MD, ductaque MN ipſi BM

107[Figure 107] perpendiculari cum ſint anguli L M N,
LDN recti, erũt anguli NMD, NDM duo-
bus rectis minores; quare MN ipſi DE oc-
curret in N; cumque angulus LMD, ęqua-
lis ſit angulo LDM, erunt reſidui ex rectis
DMN, MDN æquales, hoc eſt ND æqua-
lis NM, facto igitur centro N, interuallo
ND, deſcribatur circulus DMG, qui vtrã-
que LD, LM, continget in D, M, cum an-
guli ad D, M ſint recti: ducatur tandem
EH parallela ad DA.

Iam cum BM circulum DMG contingat
in M, ſitque ME diametro DG perpendi-
cularis erit GB ad BD, vt GE ad ED, & 1136. pri-
mi conic. permutando BG ad GE, vt BD ad DE, ſed
eſt BG maior GE, ergo & BD maior DE,
eſtque AD æqualis DC, & anguli ad verticem D æquales, quare iunctæ
BA, CE, productę, conuenient ſimul ad partes A, E, vt in F eritque FB 2271. h. BA, vt FH ad HA, & permutando BF ad FH, vt BE ad AH, vel vt BD ad
DE, vel vt BG ad GE, & diuidendo BH ad HF, vt BE ad EG, quare iuncta
FG ipſis EH, DA æquidiſtabit. Et quoniam eſt HE ad EB, vt AD ad DB,
& BE ad EM, vt BD ad DL (ob triangulorum ſimilitudinem) erit ex æquo
HE ad EM, vt AD ad DL, & quadratum HE ad quadratum EM, hoc eſt ad
rectangulum DEG, vt quadratum AD ad DL, vel vt quadratum O ad qua-
dratum Q, vel vt linea O ad P. Quod erat faciendum.

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