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348
The fifteenth Book of
PROP. V
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In an Icoſaedron given to inſcribe a Dodecaedron.
Let ABCDEF be a pyramide of the Icoſaedron, whoſe baſe is the pentagone ABCDE; and the centers of the triangles G,H,I,K,L; which connect with the right lines QH, HI, IK, KL, LG. Then GHIKL ſhall be a pentagone of the dodecaedron to be inſcribed.
For the light lines, FM, FN, FO, FP, FQ, paſſing by the centers of the triangles, aa cor. 3. 3.b 4. 1.
c 4. 1.
d 8. 1.
e 4. 1.
f 12. 13. do equally divide their baſes into two parts. b therefore the right lines MN, NO, OP, PQ, QM c are equal one to the other; d whence alſo the angles MFN, NFO, OFP, PFQ, QFM are equal. therefore the pentagone GHIKL is equiangular. e and conſequently equilateral, being FG, FH, FI, FK, FL f are equal. And if in the other eleven pyramides of the Icoſaedron, the centers of the triangles be in like ſort conjoined with right lines, then will pentagones, equal and like to the pentagone GHIKL, be deſcribed. Wherefore 12 of fuch pentagones ſhall conſtitute a dodecaedron; which alſo ſhall be deſcribed in the Icoſaedron, ſeeing the twenty angles of the dodecaedron conſiſt upon the centers of the twenty baſes of the Icoſaedron. Whereby it appears that we have deſcribed a dodecaedron in an Icoſaedron given. Which was to be done.
