Cas.2. Ponamus jam Trapezii latera oppoſita AC& BDnon
eſſe parallela. Age Bdparallelam AC& occurrentem tum rectæ
STin t,tum Conicæ ſectioni in d.Junge Cdſecantem PQin r,
& ipſi PQparallelam age DM
39[Figure 39]
ſecantem Cdin M& ABin N.
Jam ob ſimilia triangula BTt,
DBN; eſt Btſeu PQad Ttut
DNad NB.Sic & Rreſt ad
AQſeu PSut DMad AN.
Ergo, ducendo antecedentes in
antecedentes & conſequentes in
conſequentes, ut rectangulum PQ
in Rreſt ad rectangulum PSin
Tt,ita rectangulum NDMeſt
ad rectangulum ANB,& (per Caſ.1) ita rectangulum PQin Preſt
ad rectangulum PSin Pt,ac diviſim ita rectangulum PQXPR
eſt ad rectangulum PSXPT. Q.E.D.
eſſe parallela. Age Bdparallelam AC& occurrentem tum rectæ
STin t,tum Conicæ ſectioni in d.Junge Cdſecantem PQin r,
& ipſi PQparallelam age DM
39[Figure 39]ſecantem Cdin M& ABin N.
Jam ob ſimilia triangula BTt,
DBN; eſt Btſeu PQad Ttut
DNad NB.Sic & Rreſt ad
AQſeu PSut DMad AN.
Ergo, ducendo antecedentes in
antecedentes & conſequentes in
conſequentes, ut rectangulum PQ
in Rreſt ad rectangulum PSin
Tt,ita rectangulum NDMeſt
ad rectangulum ANB,& (per Caſ.1) ita rectangulum PQin Preſt
ad rectangulum PSin Pt,ac diviſim ita rectangulum PQXPR
eſt ad rectangulum PSXPT. Q.E.D.
LIBER
PRIMUS.
PRIMUS.
Cas.3. Ponamus denique lineas
40[Figure 40]
quatuor PQ, PR, PS, PTnon
eſſe parallelas lateribus AC, AB,
ſed ad ea utcunQ.E.I.clinatas. Ea
rum vice age Pq, Prparallelas
ipſi AC; & Ps, Ptparallelas
ipſi AB; & propter datos angu
los triangulorum PQq, PRr,
PSs, PTt,dabuntur rationes
PQad Pq, PRad Pr, PS
ad Ps,& PTad Pt; atque adeo rationes compoſitæ PQXPR
ad PqXPr,& PSXPTad PsXPt.Sed, per ſuperius de
monſtrata, ratio PqXPrad PsXPtdata eſt: Ergo & ratio
PQXPRad PSXPT. Q.E.D.
40[Figure 40]quatuor PQ, PR, PS, PTnon
eſſe parallelas lateribus AC, AB,
ſed ad ea utcunQ.E.I.clinatas. Ea
rum vice age Pq, Prparallelas
ipſi AC; & Ps, Ptparallelas
ipſi AB; & propter datos angu
los triangulorum PQq, PRr,
PSs, PTt,dabuntur rationes
PQad Pq, PRad Pr, PS
ad Ps,& PTad Pt; atque adeo rationes compoſitæ PQXPR
ad PqXPr,& PSXPTad PsXPt.Sed, per ſuperius de
monſtrata, ratio PqXPrad PsXPtdata eſt: Ergo & ratio
PQXPRad PSXPT. Q.E.D.
LEMMA XVIII.
Iiſdem poſitis, ſi rectangulum ductarum ad oppoſita duo latera Tra
peziiPQXPR ſit ad rectangulum ductarum ad reliqua duo late
raPSXPT in data ratione; punctumP, a quo lineæ ducuntur,
tanget Conicam ſectionem circa Trapezium deſcriptam.
peziiPQXPR ſit ad rectangulum ductarum ad reliqua duo late
raPSXPT in data ratione; punctumP, a quo lineæ ducuntur,
tanget Conicam ſectionem circa Trapezium deſcriptam.