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

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LEMMA IV. PROP. XXXIX.
Si duæ menſales ABCD, EFGH fuerint ſuper eadem linea AH
ad eaſdem partes deſcriptæ, ita vt ipſarum baſes AB, DC; EF,
HG ſint omnes inter ſe parallelæ, ſintque proportionales lateribus
in directum poſitis; nempe ſit vt AB ad EF, ita AD ad EH, & DC
ad HG. Dico, & reliqua latera BC, FG eſſe inter ſe parallela.
DVctis enim diagonalibus BD, FH, productaque BC in I. Cum ſit BA
ad EF, vt AD ad EH, erit permutando BA ad AD, vt FE ad EH, eſtq;
angulus BAD ęqualis angulo FEH, ob parallelas BA, FE, quare triangulum
BAD ſimile eſt triangulo FEH; ideoq; angulus BDA æqualis angulo FHE,
& totus CD A, æquatur
62[Figure 62] toto GHE, ob parallelas
CD, GH, vnde reliquus
CDB, æquatur reliquo
GHF. Item cum ſit CD
ad GH, vt DA ad HE,
erit permutando CD ad
DA, vt GH ad HE, eſtq;
DA ad DB, vt HE ad HF,
ob triangulorum ADB,
EHF ſimilitudinem, qua-
re ex æquali CD ad DB,
erit vt GH ad HF, ſuntq; anguli ad D, & H æquales, ergo, & angulus D CB,
ſiue HIB, æqualis angulo HGF, ideoque rectæ BC, FG inter ſe æquidiſtant.
Quod erat, & c.

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