Page:Euclid's Elements 1714 Barrow translation.djvu/360

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In an Icoaedron given to incribe a Dodecaedron.

Let ABCDEF be a pyramide of the Icoaedron, whoe bae 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 hall be a pentagone of the dodecaedron to be incribed.

For the light lines, FM, FN, FO, FP, FQ, paing by the centers of the triangles, a do equally divide their baes into two parts. b therefore the right lines MN, NO, OP, PQ, QM c are equal one to the other; d whence alo the angles MFN, NFO, OFP, PFQ, QFM are equal. therefore the pentagone GHIKL is equiangular. e and conequently equilateral, being FG, FH, FI, FK, FL f are equal. And if in the other eleven pyramides of the Icoaedron, the centers of the triangles be in like ort conjoined with right lines, then will pentagones, equal and like to the pentagone GHIKL, be decribed. Wherefore 12 of fuch pentagones hall contitute a dodecaedron; which alo hall be decribed in the Icoaedron, eeing the twenty angles of the dodecaedron conit upon the centers of the twenty baes of the Icoaedron. Whereby it appears that we have decribed a dodecaedron in an Icoaedron given. Which was to be done.