166146GEOMETRIÆ
FG, HM, homologæ figurarum, FGP, HMQ, quę figurę vo-
cantur in ipſis ſolidis homologæ. Dico hæc ſolida eſſe proportiona-
lia; ſit duarum, AB, CD, tertia proportionalis, R, quarta, S, &
duarum, FG, HM, tertia, I, quarta, T, eſt igitur ſolidum, AXB,
ad, CVD, vt, AB, ad, S, . i. vt, FG, ad, T, (quia vt, AB, ad,
11Ex Corol.
2. antec. CD, ita eſt, FG, ad, HM,) . i. vt ſolidum, FOGP, ad, HNM
Q, quod eſt propoſitum.
cantur in ipſis ſolidis homologæ. Dico hæc ſolida eſſe proportiona-
lia; ſit duarum, AB, CD, tertia proportionalis, R, quarta, S, &
duarum, FG, HM, tertia, I, quarta, T, eſt igitur ſolidum, AXB,
ad, CVD, vt, AB, ad, S, . i. vt, FG, ad, T, (quia vt, AB, ad,
11Ex Corol.
2. antec. CD, ita eſt, FG, ad, HM,) . i. vt ſolidum, FOGP, ad, HNM
Q, quod eſt propoſitum.
Sit nunc ſolidum, AXB, ad ſibi ſimile, CVD, vt, FOGP, ad
95[Figure 95]
ſibi ſimile, HNMQ, &
ſint ea-
dem deſcribentes, AB, CD, li-
neæ, vel latera homologa figura-
rum homologarum, AEBY, C
KD ℟, & , FG, HM, duo po
ſtrema deſcribentes ſint lineæ, vel
latera homologa figurarum ho-
mologarum, FGP, HMQ. Di-
co has eſſe proportionales; ſint
adhuc duarum, AB, CD, tertia
proportionalis, R, quarta, S, &
duarum, FG, HM, tertia, I,
quarta, T, quia ergo ſolida, AX
B, CVD, ſunt ſimilia erit, AXB, ad, CVD, vt, AB, ad, S, & ,
22Ex Corol.
2. antec. FOGP, ad, HNMP, vt, FG, ad, T, ſunt autem hæc quatuor
ſolida proportionalia, ergo & , AB, ad, S, erit vt, FG, ad, T, er-
go, AB, ad, CD, erit vt, FG, ad, HM, quod oſtendendum erat.
dem deſcribentes, AB, CD, li-
neæ, vel latera homologa figura-
rum homologarum, AEBY, C
KD ℟, & , FG, HM, duo po
ſtrema deſcribentes ſint lineæ, vel
latera homologa figurarum ho-
mologarum, FGP, HMQ. Di-
co has eſſe proportionales; ſint
adhuc duarum, AB, CD, tertia
proportionalis, R, quarta, S, &
duarum, FG, HM, tertia, I,
quarta, T, quia ergo ſolida, AX
B, CVD, ſunt ſimilia erit, AXB, ad, CVD, vt, AB, ad, S, & ,
22Ex Corol.
2. antec. FOGP, ad, HNMP, vt, FG, ad, T, ſunt autem hæc quatuor
ſolida proportionalia, ergo & , AB, ad, S, erit vt, FG, ad, T, er-
go, AB, ad, CD, erit vt, FG, ad, HM, quod oſtendendum erat.
THE OREMA XIX. PROPOS. XIX.
SI in parallelogrammo diameter ducta fuerit, parallelo-
grammum duplum eſt cuiuſuis triangulorum per ipſam
diametrum conſtitutorum.
grammum duplum eſt cuiuſuis triangulorum per ipſam
diametrum conſtitutorum.
Sit parallelogrammum vtcunque, AD, in
96[Figure 96]
quo ducta diameter, FC, ipſum diuidat in tri-
angula, FAC, CDF. Dico parallelogram-
mum, AD, duplum eſſe cuiuſuis triangulo-
rum, FAC, CDF; abſcindanturab, FD, C
A, verſus puncta, F, C, partes æquales, FE,
CB, & per puncta, B, E, parallelæ ipſi baſi,
CD, ducantur, EH, BM, incidentes diame-
tro, FC, in punctis, H, M; quoniam ergo in triangulis, FHE,
angula, FAC, CDF. Dico parallelogram-
mum, AD, duplum eſſe cuiuſuis triangulo-
rum, FAC, CDF; abſcindanturab, FD, C
A, verſus puncta, F, C, partes æquales, FE,
CB, & per puncta, B, E, parallelæ ipſi baſi,
CD, ducantur, EH, BM, incidentes diame-
tro, FC, in punctis, H, M; quoniam ergo in triangulis, FHE,