PROPOSITIO XXIV.
Si duarum pyramidum triangul as baſes haben
tium æqualium, & ſimilium inter ſe, tria latera
tribus lateribus homologis fuerint in directum
conſtituta, in vertice communi erit vtriuſque ſi
mul centrum grauitatis.
tium æqualium, & ſimilium inter ſe, tria latera
tribus lateribus homologis fuerint in directum
conſtituta, in vertice communi erit vtriuſque ſi
mul centrum grauitatis.
Sint duæ pyramides ſimiles, & æquales, quarum ver
tex communis G, baſes autem triangula ABC, DEF.
Et ſint latera homologa pyramidum in directum inter ſe
conſtituta: vt AG, GF: & BG, GD, & CG, GE.
Dico compoſiti ex duabus pyramidibus ABCG, GDEF,
ita conſtitut is centrum gra
uitatis eſse in puncto G.
Eſto enim H, centrum gra
uitatis pyramidis ABCG,
& ducta HGK, ponatur
GK, æqualis GH, & iun
gantur EK, KD, BH,
CH. Quoniam igitur eſt
vt HG, ad GK, ita CG,
ad GE, & proportio eſt
æqualitatis: & angulus
HGC, æqualis angulo EG
K, erit triangulum CGH,
35[Figure 35]
ſimile, & æquale triangulo EGK. Similiter triangulum
BGH, trian gulo DGK; & triangulum BGC, triangu
lo DGE: quare & triangulum BCH, triangulo DEK.
pyramis igitur BCGH, ſimilis, & æqualis eſt pyramidi
EDGK. Congruentibus igitur inter ſe duobus triangu
tex communis G, baſes autem triangula ABC, DEF.
Et ſint latera homologa pyramidum in directum inter ſe
conſtituta: vt AG, GF: & BG, GD, & CG, GE.
Dico compoſiti ex duabus pyramidibus ABCG, GDEF,
ita conſtitut is centrum gra
uitatis eſse in puncto G.
Eſto enim H, centrum gra
uitatis pyramidis ABCG,
& ducta HGK, ponatur
GK, æqualis GH, & iun
gantur EK, KD, BH,
CH. Quoniam igitur eſt
vt HG, ad GK, ita CG,
ad GE, & proportio eſt
æqualitatis: & angulus
HGC, æqualis angulo EG
K, erit triangulum CGH,
35[Figure 35]ſimile, & æquale triangulo EGK. Similiter triangulum
BGH, trian gulo DGK; & triangulum BGC, triangu
lo DGE: quare & triangulum BCH, triangulo DEK.
pyramis igitur BCGH, ſimilis, & æqualis eſt pyramidi
EDGK. Congruentibus igitur inter ſe duobus triangu