The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

The 28. Theoreme. The 38. Proposition. Triangles which consist vppon equall bases, and in the selfe same parallel lines, are equall the one to the other.

SVppose that these triangles ABC and DEF do consist vpon equal ba∣ses, that is, vpon BC and EF, and in the selfe same parallel lines, that is BF and AD. Then I say that the triangle ABC is equall to the trian∣gle

ABC is equall to the triangle DEF.* 1.1 Produce (by the second peticion) the line AD on eche side to the pointes G and H. And (by the 31. proposition) by the point B drawe vnto CA a paral∣lel

[illustration]

line BG, and (by the same) by the pointe F drawe vnto DE a parallel line FH.* 1.2 VVherfore GBCA and DEFH are parallelogrammes. But the parallelograme GBCA is (by the 36 proposition) equal to the paralle∣logramme DEFH, for they consist vpon equall bases, that is, BC and EF, and are in the selfe same parallel lines, that is, BF and GH. But (by the 34. proposition) the triangle ABC is the halfe of the parallelogramme GBCA, for the diameter AB deui∣deth it into two equall partes: and the triangle DEF is (by the same) the halfe of the parallelogramme DEFH, for the diameter FD deuideth it into two e∣quall partes. But the halues of thinges equall are (by the 7. common sentence) e∣quall ••he one to the other. VVherfore the triangle ABC is equall to the trian∣gle DEF. VVherefore triangles which consist vppon equall bases, and in the selfe same parallel lines, are equall the one to the other: which was required to be proued.

In this proposition are three cases.* 1.3 For the bases of the triangles either haue one part common to them both or the base of the one toucheth the base of the other onely in a point: or their bases are vtterly seuered a sunder. And ech of these cases may also be diuersly,* 1.4 as we before haue sene in parallelogrammes con¦sisting on equall bases, and being in the selfe same parallel lines. So that he which diligently noteth the variety that was there put touching them, may also easely frame the same varietie to ech case in this proposition. VVherefore I thinke it nedeles here to repeate the same agayne: for how soeuer the bases be put, or the toppes, the manner of construction and demonstration here put by Euclide will serue: namely, to draw parallel lines to the sides.

An addition of Pelitarius.

To deuide a triangle geuen into two equall partes.

Suppose that the triangle geuen to be deuided in to two

[illustration]

equall partes be ABC.* 1.5 Deuide one of the sides therof, namely, BC into two equall partes (by the 10. propo∣sition) in the point D. And draw a line from the point D to the point A. Thē I say that the two triangles ABD & ACD, are equal, which is easy to proue (by the 38. pro∣position) if by the point A we drawe vnto the line BC a paral••el line (by the 31. proposition), which let by HK: for so the triangles AB D and ADC, consisting vppon equal bases BD & DC, and being in the selfe same paral∣lel lines HK and BC are of necessitie equall. The selfe

same thing also wil happen if the side BA be deuided into two equall parts in the point E, and so be drawen a right line from the point E, to the point C. Or if the side AC be deuided into two equall partes in the point F, and so be drawen a right line from the point F to the point B: which is in like manner proued by drawing parallel lines by the pointes B, and C, to the lines BA and AC,

* 1.6And so by this you may deuide any triangle into so many partes as are sig∣nified by any number that is euenly euen: as into 14.16.32.64. &c.

An other addition of Pelitarius.* 1.7

From any point geuen in one of the sides of a triangle, to draw a line which shal deuide the trian∣gle into two equall partes.

Let the triangle geuen be BCD: and let the point geuen in the side BC be A. It is required from the point A to draw a line which shal deuide the triangle BCD into two equall partes.* 1.8 Deuide the side BC into two equall partes in the point E. And drawe a right line from the point A to the point D. And (by the

[illustration]

31. proposition) by the point E draw vnto the line AD a parallel line EF: which let cutte the side DC in the point F. And draw a line from the point A to the point F. Then I say that the line AF deuideth the triangle BCD into two equall partes: namely, the trapesium ABDF is equall to the triangle ACF. For draw a line from E to D,* 1.9 cutting the line AF in the point G. Now then it is manifest (by the 38. proposition) that the two trian∣gles BED and CED are equall (if we vnderstand a line to be drawen by the point D parallel to the line BC, for the bases BE and EC are equal). The two triangles also DEF and AEF are (by the 37. proposition) equall: for they consist vpon one and the selfe same base EF, and are in the selfe same parallel lines AD and EF. Wherefore taking away the triangle EFG which is cōmō to thē both, the triangle AEG shalbe equall to the triangle DFG: wher¦fore vnto either of thē adde the trapesiū CFGE, and the triangle ACF shalbe equal to the triangle DEC. But the triangle DEC is the halfe part of the whole triangle BCD wherefore the triangle ACF is the halfe part of the same triangle BCD. Wherfore the residue, namely, the trapesium ABFD is the other halfe of the same triangle. Where∣fore the line AF deuideth the whole triangle BCD into two equall partes: which was required to be done.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstration.
• * 1.3

  Thre cases in this proposition.
• * 1.4

  Ech of these ca∣ses also may be diuersly.
• * 1.5

  An addition of Pelitarius, to deuide a trian∣gle into two e∣quall partes.
• * 1.6

  Note.
• * 1.7

  An other addi∣tion of Pelita∣rius.
• * 1.8

  Construction.
• * 1.9

  Demonstration