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 6. Theoreme. The 6. Proposition. If there be two triangles wherof the one hath one angle equall to one angle of the other, & the sides including the equall an∣gles be proportionall: the triangles shall be equiangle, and those angles in them shall be equall, vnder which are subten∣ded sides of like proportion.

SVppose that there be two triangles ABC, and DEF, which let haue the angle BAC of the one triangle equall vnto the angle EDF of the other triangle, and let the sides including the equall angles be proportio∣nall, that is, as BA is to AC, so let ED be to DF. Then I say, that y^e triangle ABC is equiangle vnto the triangle DEF: and the angle ABC is equall vn∣to the angle DEF, and the angle ACB equall vnto the angle DFE, which ••ngles are subtēded to sides of like pro∣portion.

[illustration]

Vnto the right line DF, and to the poynt in it D (by the 23. of the first) describe vnto either of y^e angles BAC and EDF,* 1.1 an equall angle FDG. And vnto the right line DF, and vnto the point in it F (by y^e same) describe vnto y^e angle AC•• an equall angle DFG. And forasmuch as the two angles BAC and ACB, are (by the 17. of the first) lesse then two right angles: therefore also the two angles EDG and DFG, are lesse then two right angles. Wherfore y^e lines DG & FG being produced, shall cōcurre (by the 5. petition). Let thē concurre in the point G. Wherefore DFG is a triangle.

Wherefore the angle remaining ABC is equall vnto the angle remaining DGF (by the 32. of the first). Wherefore the triangle ABC is equiangle vnto the tri∣angle DGF. Wherefore as BA is in proportion to AC, so is GD to DF (by the 4. of the sixt). But it is supposed, that as BA is to AC, so is ED to DF. Wherefore (by the 11. of the fift) as ED is to DF, so is GD to DF. Where∣fore (by the 9. of the fift) ED is equall vnto DG. And DF is common vnto them both. Now then there are two

[illustration]

sides ED and DF equall vnto two sides GD and DF: and the angle EDF (by supposition) is equall vnto the angle GDF. Wherefore (by the 4. of the first) the base EF is equall vnto the base GF, and the triangle DEF is (by the same) equall vnto the triangle GDF, and the other an∣gles remayning in them are equall the one to the other, vnder which are sub∣tended equall sides. Wherefore the an∣gle DFG is equall vnto the angle DFE: and the angle DGF vnto the angle DEF. But the angle DFG is (by construction) equall vnto y^e angle ACB. And the angle DGF is as it hath bene proued, equall to y^e angle ABC. Wherfore also y^e angle ACB is equall vnto the angle DFE. And y^e angle ABC is equall to the angle DEF. But by sup∣position the angle BAC is equall vnto the angle EDF. Wherefore the tri∣angle ABC is equiangle vnto the triangle DEF. If therefore there be two triangles, whereof the one hath one angle equall to one angle of the other, and if also the sides including the equall angles be proportionall: then shall the triangles also be equiangle, and those angles in them shalbe equall, vnder which are subtended sides of like proportion: which was required to be proued.

Notes

• * 1.1

  Constructio••.