272252GEOMETRIÆ
BOMFH, erunt vt, AN, ad figuram, BDMO, quod oſtendere
oportebat. Per hanc autem, & antecedentem Propoſit. vniuerſa-
lius oſtenduntur Propoſ. 15. 16. necnon Corollaria Prop. 19. & 20.
oportebat. Per hanc autem, & antecedentem Propoſit. vniuerſa-
lius oſtenduntur Propoſ. 15. 16. necnon Corollaria Prop. 19. & 20.
THEOREMA XXX. PROPOS. XXXI.
SI parallelogrammum fuerit ellipſi circumſcriptum, ita ta-
men, vt eiuſdem latera non tangant ellipſim in extremis
punctis axium eiuſdem; portiones coalternè tangentes erunt
æquales; & ſi duabus oppoſitis tangentibus ducantur paral-
lelę abſcindentes à reliquis coalternis tangentibus rectas li-
neas æquales, ſumptas verſus puncta contactuum; rectangu-
lum, quod continetur ſub vnius parallelarum ea parte, quæ
manet intra curuam ellipſis, & tangentem ex ea parte, & ſub
reliqua illi in directum manente intra ellipſim, erit æquale
rectangulo ad coalternam tangentem ſimiliter ſumpto.
men, vt eiuſdem latera non tangant ellipſim in extremis
punctis axium eiuſdem; portiones coalternè tangentes erunt
æquales; & ſi duabus oppoſitis tangentibus ducantur paral-
lelę abſcindentes à reliquis coalternis tangentibus rectas li-
neas æquales, ſumptas verſus puncta contactuum; rectangu-
lum, quod continetur ſub vnius parallelarum ea parte, quæ
manet intra curuam ellipſis, & tangentem ex ea parte, & ſub
reliqua illi in directum manente intra ellipſim, erit æquale
rectangulo ad coalternam tangentem ſimiliter ſumpto.
Sit ergo ellipſis, BDMG, cui ſit circumſcriptum parallelogram-
mum, AR, ita tamen, vt puncta contactuum non ſint puncta ex-
trema axium eiuſdem, tangant autem in punctis, BDMG, & iun-
gantur, BM, DG, & quoniam, AC, FR, ſunt tangentes paralle-
læ, vt etiam, AF, CR, ideò, BM, GD, per centrum ellipſis tran-
11Elicitur
ex 27. 2.
Con. ſibunt, ſit earum communis ſectio punctum, S, ergo, S, erit centrum
169[Figure 169]
ellipſis, cum, BM, GD, ſint diametri.
Dico ergo portiones laterum parallelo-
grammi, AR, coalternè tangentes eſſe
æquales. ſ. AD, ipſi, GR, AB, ipſi, M
R, BC, ipſi, FM, & , CG, ipſi, DF;
iungantur, BG, DM; in triangulis ergo,
BSG, DSM, latus, BS, æquatur late-
ri, SM, & latus, GS, lateri, SD, item
angulus; BSG, angulo, DSM, ergo ba-
224. 1. Elem. ſis, BG, æquatur baſi, DM, & angulus,
SBG, angulo, SMD, & , SGB, ipſi, S
DM, totus autem angulus, CBS, æquatur toti, FMS, ſibi coal-
terno, ergo reliquus angulus, CBG, æquatur reliquo angulo, DM
F, & ſimiliter probabimus angulum, BGC, æquari angulo, MD
F, ergo reliquus, BCG, æquabitur reliquo, DFM, (qui etiam ſunt
æquales, quia ſunt anguli oppoſiti parallelogrammi, AR,) & ideò
trianguli, BCG, DFM, erunt æquianguli, & , BG, DM,
mum, AR, ita tamen, vt puncta contactuum non ſint puncta ex-
trema axium eiuſdem, tangant autem in punctis, BDMG, & iun-
gantur, BM, DG, & quoniam, AC, FR, ſunt tangentes paralle-
læ, vt etiam, AF, CR, ideò, BM, GD, per centrum ellipſis tran-
11Elicitur
ex 27. 2.
Con. ſibunt, ſit earum communis ſectio punctum, S, ergo, S, erit centrum
Dico ergo portiones laterum parallelo-
grammi, AR, coalternè tangentes eſſe
æquales. ſ. AD, ipſi, GR, AB, ipſi, M
R, BC, ipſi, FM, & , CG, ipſi, DF;
iungantur, BG, DM; in triangulis ergo,
BSG, DSM, latus, BS, æquatur late-
ri, SM, & latus, GS, lateri, SD, item
angulus; BSG, angulo, DSM, ergo ba-
224. 1. Elem. ſis, BG, æquatur baſi, DM, & angulus,
SBG, angulo, SMD, & , SGB, ipſi, S
DM, totus autem angulus, CBS, æquatur toti, FMS, ſibi coal-
terno, ergo reliquus angulus, CBG, æquatur reliquo angulo, DM
F, & ſimiliter probabimus angulum, BGC, æquari angulo, MD
F, ergo reliquus, BCG, æquabitur reliquo, DFM, (qui etiam ſunt
æquales, quia ſunt anguli oppoſiti parallelogrammi, AR,) & ideò
trianguli, BCG, DFM, erunt æquianguli, & , BG, DM,
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