274254GEOMETRIÆ
COROLLARIVM.
_H_Inc patet nedum rectangulum, NOV, æquari rectangulo, IHP,
ſed etiam portiones interceptas tangentibus, & curuaellipſis eſſe
inter ſe æquales, belut, OV, ipſi, PH.
ſed etiam portiones interceptas tangentibus, & curuaellipſis eſſe
inter ſe æquales, belut, OV, ipſi, PH.
THEOREMA XXXI. PROPOS. XXXII.
EXpoſita ellipſi, cum parallelogrammo illi circumſcripto
Theor. antecedentis, cæteris omiſſis, oſtendemus, re-
gula, FR, omnia quadrata parallelogrammi, AR, ad om-
nia quadrata ellipſis, BDMG, cum rectangulis bis ſub ea-
dem ellipſi, & ſub trilineis, BCG, GRM, eſſe, vt paralle-
logrammum, AR, ad ellipſim, BDMG.
Theor. antecedentis, cæteris omiſſis, oſtendemus, re-
gula, FR, omnia quadrata parallelogrammi, AR, ad om-
nia quadrata ellipſis, BDMG, cum rectangulis bis ſub ea-
dem ellipſi, & ſub trilineis, BCG, GRM, eſſe, vt paralle-
logrammum, AR, ad ellipſim, BDMG.
Ducantur à punctis contactuum regulæ, FR, parallelę, GE, D
11Coroll. 1.
26. lib. 2. V; omnia ergo quadrata, AR, ad rectangula ſub ellipſi, BDMG,
& ſub, AR, ſunt vt, AR, ad ellipſim, BDMG; verum rectangula
ſub ellipſi, BDMG, & ſub, AR, ſunt æqualia rectangulis ſub el-
22A. 23. 1. 2. lipſi, BDMG, & ſub duobus trilineis, BAD, DFM, item ſub el-
lipſi, BDMG, & ſub eadem. i. omnibus quadratis ellipſis, BDM
G, & ſub eadem ellipſi, BDMG, & ſub duobus trilineis, BCG,
GRM, verum rectangula ſub ellipſi, BDMG, & ſub trilineis, B
170[Figure 170]
AD, DFM, æquantur rectangulis ſub ea-
dem ellipſi, & ſub trilineis, BCG, GRM,
quod ſic patet, quoniam enim, AD, RG,
33Exantec. coalternè tangentes ſunt æquales, & ductis
ipſi, FR, parallelis intra ellipſim, ex ipſis
coalternè tangentibus, AD, RG, abſcin
dentibus portiones æquales verſus puncta
contactuum, rectangula ſumpta, vt dictum
eſt in antecedenti Theor. ſunt æqualia, ideò
& omnia omnibus erunt æqualia. ſ. rectan-
gula ſub portione, OGBD, & trilineo, BAD, erunt æqualia re-
ctangulis ſub portione, SMG, & ſub trilineo, GMR, eadem ra-
tione rectangula ſub portione, OMD, & trilineo, DFM, æquan-
tur rectangulis ſub portione, SBG, & trilineo, BCG, ergo rectan-
gula ſub ellipſi, BDMG, & duobus trilineis, BAD, DFM, ęquan-
tur rectangulis ſub ellipſi, BDMG, & ſub trilineis, BCG, GRM;
11Coroll. 1.
26. lib. 2. V; omnia ergo quadrata, AR, ad rectangula ſub ellipſi, BDMG,
& ſub, AR, ſunt vt, AR, ad ellipſim, BDMG; verum rectangula
ſub ellipſi, BDMG, & ſub, AR, ſunt æqualia rectangulis ſub el-
22A. 23. 1. 2. lipſi, BDMG, & ſub duobus trilineis, BAD, DFM, item ſub el-
lipſi, BDMG, & ſub eadem. i. omnibus quadratis ellipſis, BDM
G, & ſub eadem ellipſi, BDMG, & ſub duobus trilineis, BCG,
GRM, verum rectangula ſub ellipſi, BDMG, & ſub trilineis, B
dem ellipſi, & ſub trilineis, BCG, GRM,
quod ſic patet, quoniam enim, AD, RG,
33Exantec. coalternè tangentes ſunt æquales, & ductis
ipſi, FR, parallelis intra ellipſim, ex ipſis
coalternè tangentibus, AD, RG, abſcin
dentibus portiones æquales verſus puncta
contactuum, rectangula ſumpta, vt dictum
eſt in antecedenti Theor. ſunt æqualia, ideò
& omnia omnibus erunt æqualia. ſ. rectan-
gula ſub portione, OGBD, & trilineo, BAD, erunt æqualia re-
ctangulis ſub portione, SMG, & ſub trilineo, GMR, eadem ra-
tione rectangula ſub portione, OMD, & trilineo, DFM, æquan-
tur rectangulis ſub portione, SBG, & trilineo, BCG, ergo rectan-
gula ſub ellipſi, BDMG, & duobus trilineis, BAD, DFM, ęquan-
tur rectangulis ſub ellipſi, BDMG, & ſub trilineis, BCG, GRM;