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 2. Theoreme. The 2. Proposition. If to any one of the sides of a triangle be drawen a parallel

right line, it shall cut the sides of the same triangle proportio∣nally. And if the sides of a triangle be cut proportionally, a right lyne drawn from section to section is a parallel to the o∣ther side of the triangle.

SVppose that there be a triangle ABC, vnto one of the sides whereof, namely, vnto BC, let there be drawen a parallel line DE cuttyng the sides AC and AB in the pointes E and D. Then I say first that as BD is to DA, so is CE to EA.* 1.1 Draw a line from B to E, & also from C to D. Wher¦fore (by the 37. of the first) the triangle BDE is equall vnto the triangle CDE: for they are set vpon one and the same base DE, and are contained within the selfe same parallels DE and BC. Consider

[illustration]

also a certaine other triangle ADE. Now thinges equall (by the 7. of the fifth) haue to one selfe thing one and the same proportion. Wherfore as y^e triangle BDE is to y^e triangle ADE, so is y^e triangle CDE to the triangle ADE. But as y^e triangle BDE is to y^e trian∣gle ADE, so is y^e base BD to y^e base DA (by the first of this booke.) For they are vnder one and the selfe same toppe, namely, E, and ther∣fore are vnder one and the same altitude. And by the same reason as the triangle CDE is to the triangle ADE, so is the lyne CE to the lyne EA. Wherfore (by the 11. of the fifth) as the line BD is to the line DA, so is the line CE to the line EA.

But now suppose that in y^e triangle ABC the sides AB & AC be cut propor∣tionally so y^t as BD is to DA, so let CE be to EA, & draw a line from D to E. Then secondly I say y^t the line DE is a parallel to y^e lyne BC.* 1.2 For the same order of construction being kept, for y^t as BD is to DA, so is CE to EA, but as BD is to DA, so is y^e triangle BDE to y^e triangle ADE (by the 1. of the sixt) & as CE is to EA, so (by y^e same) is the triangle CDE to y^e triangle ADE: therfore (by the 11. of the fifth) as the triangle BDE is to the triangle ADE, so is the triangle CDE to the triangle ADE. Wherfore either of these triangles BDE and CDE haue to the triangle ADE one and the same proportion. Wherefore (by the 9. of the fifth) the triangle BDE is equall vnto the triangle CDE, and they are vpon one and the selfe base, namely, DE. But triangles equall and set vpon one base, are also contained within the same parallel lines (by the 39. of the first.) Wherfore the line DE is vnto the line BC a parallel. If therfore to a∣ny one of the sides of a triangle be drawn a parallel line, it cutteth the other sides of the same triangle proportionally. And if the sides of a triangle be cut propor∣tionally, a right lyne drawen from section to section, is parallel to the other side of the triangle: which thing was required to be demonstrated.

¶ Here also Flussates addeth a Corollary.

If a line parallel to one of the sides of a triangle do cut the triangle, it shall cut of from the whole triangle a triangle like to the whole triangle.* 1.3 For as it hath bene proued it deui∣deth the sides proportionally. So that as EC is to EA, so is BD to DA, wherfore by the 18. of the fifth, as AC is to AE, so is AB to AD. Wherfore alternately by the 16. of the fifth as AC is to AB, so is AE to AD: wherefore in the two trian∣gles EAD and CAB the sides about the common angle A are proportional. The sayd triangles also are equiangle. For forasmuch as the right lynes AEC and ADB do fall vpon the parallel lynes ED and CB, therefore by the 29. of the firs•• they make the angles AED and ADE in the triangle ADE equall to the angles ACB and ABC in the triangle ACB. Wherefore by the first definition of this booke the whole triangle ABC is like vnto the triangle cut of ADE.

Notes

• * 1.1

  The first part of this Theo∣reme.
• * 1.2

  Demonstra∣tion of the second part.
• * 1.3

  A Corollary added by Flussates.