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

An other addition of Pelitarius.* 1.1

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.* 1.2 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,* 1.3 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

  An other addi∣tion of Pelita∣rius.
• * 1.2

  Construction.
• * 1.3

  Demonstration