The Way to Geometry, Book 7
Of Geometry, the seventh Booke, Of the comparison of Triangles.
1 Equilater triangles are equiangles. 8. p.j.
Thus forre of the Geometry, or affections and reason of one triangle; the comparison of two triangles one with another doth follow. And first of their rate or reason, out of their sides and angles: Whereupon triangles betweene themselves are said to be equilaters and equiangles. First out of the equality of the sides, is drawne also the equalitie of the angles.
Triangles therefore are here jointly called equilaters, whose sides are severally equall, the first to the first, the second, to the second, the third to the third; although every severall triangle be inequilaterall. Therefore the equality of the sides doth argue the equality of the angles, by the 7. e iij. As here.
2 If two triangles be equall in angles, either the two equicrurals, or two of equall either shanke, or base of two angles, they are equilaters, 4. and 26. p j.
Oh thus; If two triangles be equall in their angles, either in two angles contained under equall feet, or in two angles, whose side or base of both is equall, those angles are equilater. H.
This element hath three parts, or it doth conclude two triangles to be equilaters three wayes. 1. The first part is apparent thus: Let the two triangles be a e i, and o u y; because the equall angles at a, and o, are equicrurall, therefore they are equall in base, by the 7. e iij.
2 The second thus: Let the said two triangles a e i, and o u y, be equall in two angles a peece, at e, and i, and at u, and y. And let them be equall in the shanke e i, to u y. I say, they are equilaters. For if the side a e, (for examples sake) be great or than the side o u, let e s, be cut off equall unto it; and draw the right line i s. Here by the antecedent, the triangles s e i, and o u y, shall be equiangles, and the angles s i e, shall be equall to the angle o y u, to which also the whole angle a i e, is equall, by the grant. Therefore the whole and the part are equall, which is impossible. Wherefore the side a e, is not unequall but equall to the side o u: And by the antecedent or former part, the triangles a e i, and o u y, being equicrurall, are equall, at the angle of the shanks: Therefore also they are equall in their bases a i, and o y.
3 The third part is thus forced: In the triangles a e i, and o u y, let the angles at e, and i, and u, and y, be equall, as afore: And a e. the base of the angle at i, be equall to o u, the base of angle at y: I say that the two triangles given are equilaters. For if the side e i, be greater than the side u y, let e s, be cut off equall to it, and draw the right line a s. Therefore by the antecedent, the two triangles, a e s, and o u y, equall in the angle of their equall shankes are equiangle; And the angle a s e, is equall to the angle o y u, which is equall by the grant unto the angle a i e. Therefore a s e, is equall to a i e, the outter to the inner, contrary to the 15. e v. j. Therefore the base e i, is not unequall to the base u y, but equall. And therefore as above was said, the two triangles a e i, and o u y, equall in the angle of their equall shankes, are equilaters.
3. Triangles are equall in their three angles.
The reason is, because three angles in any triangle are equall to two right angles, by the 13. e v j. As here, the greatest triangle, all his corners joyntly taken, is equall to the least.
And yet notwithstanding it is not therefore to be thought to be equiangle to it: For Triangles are then equiangles, when the severall angles of the one, are equall to the severall angles of the other: Not when all joyntly are equall to all.
Therefore
4. If two angles of two triangles given be equall, the other also are equall.
All the three angles, are equall betweene themselves… by the 3 e. Therefore if from equall you take away equall, those which shall remaine shall be equall.
5. If a right triangle equicrurall to a triangle be greater in base, it is greater in angle: And contrariwise. 25. and 24. pj.
Thus farre of the reason or rate of equality, in the sides and angles of triangles: The reason of inequality, taken out of the common and generall inequality of angles, doth follow. The first is manifest, by the 9 eiij. as here thou seest in a e i, and o u y.
6. If a triangle placed upon the same base, with another triangle, be lesser in the inner shankes, it is greater in the angle of the shankes.
This is a consectary drawne also out of the 10 e iij. As here in the triangle a e i, and a o i, within it and upon the same base. Or thus: If a triangle placed upon the same ba…e with another triangle, be lesse then the other triangle, in regard of his feet, (those feete being conteined within the feete of the other triangle) in regard of the angle conteined under those feete, it is greater: H.
7. Triangles of equall heighth, are one to another as their bases are one to another.
Thus farre of the Reason or rate of triangles: The proportion of triangles doth follow; And first of a right line with the bases. It is a consectary out of the 16 e iiij.
Therefore
8. VponUpon an equall base, they are equall.
This was a generall consectary at the 16 e iiij: From whence Archimedes concluded, If a triangle of equall heighth with many other triangles, have his base equall to the bases of them all, it is equall to them all: as here thou seest a e i to be equall to the triangles a e o, u o y, s y r, l r m, n m i. Here hence also thou mayst conclude, that Equilater triangles are equall; Because they are of equall heighth, and upon the same base.
9. If a right line drawne from the toppe of a triangle, doe cut the base into two equall parts, it doth also cut the triangle into two equall parts: and it is the diameter of the triangle.
As here thou seest: For the bisegments, or two equall portions thus cut are two triangles of equall heighth (that that is to say, they have one toppe common to both, within the same parallels) and upon equall bases: Therefore they are equall: And that right line shall be the diameter of the triangle, by the 5 e iiij, because it passeth by the center.
10. If a right line be drawne from the toppe of a triangle, unto a point given in the base (so it be not in the middest of it) and a parallell be drawne from the middest of the base unto the side, a right line drawne from the toppe of the sayd parallell unto the sayd point, shall cut the triangle into two equall parts.
Let the triangle given be a e i: And let a o, cut the base e i, in o unequally: And let u y be parallell from u, the middest of y base, unto the sayd e i. I say that y o shall divide the triangle into two equall portions. For let a u be knit together with a right line: That line, by the 9 e, shall divide the triangle into two equall parts. Now the two triangles a y u, and y o u, are equall by the 8 e; because they are of equall height, and upon the same base. Take away y s u, the common triangle; And you shall leave a s y, and o s u, equall betweene themselves: The common right lined figure y s u i, let it be added to both the sayd equall triangles: And then … y i, shall be equall to a u i, the halfe part: And therefore a e o y, the other right lined figure, shall be the halfe of the triangle given.
11 If equiangled triangles be reciprocall in the shankes of the equall angle, they are equall: And contrariwise. 15. p. vj. Or thus, as the learned M…. Brigges hath conceived it: If two triangles, having one angle, are reciprocall, &c.
Direct proportion in triangles, is such as hath in the former beene taught: Reciprocall proportion followeth. It is a consectary drawne out of the 18 e iiij; which is manifest, as oft as the equall angle is a right angle: For then those shankes, [comprehending the equall angles,] are the heights and the bases: As here thou seest in the severed triangles. Notwithstanding in obliquangle triangles, although the shankes are not the heights, the cause of the truth hereof is the same. Yet if any man shall desire a demonstration of it, it is thus: Let therefore the diagramme or figure bee in the triangles a e i, and a o u: And the angles o a u, and e a i, let them be equall: And as u a is to a e, so let i a be unto a o: I say that the triangles a o u, and e a i, are equall. For e o being knit together with a right line, u a o is unto o a e, as u a is unto a e, by the 7 e: And i a, unto a o, by the grant, is as e a i is unto e a o. Therefore u a o, and e a i, are unto e a o proportionall: And therefore they are equall one to another.
The converse, is concluded by the same sorites, but by saying all backward. For u a unto a e is, as u a o is unto o a e, by the 7 e: And as e a i, by the grant: Because they are equall: And as i a is unto a o, by the same. Wherefore u a is unto a e, as i a is unto a o.
12 If two triangles be equiangles, they are proportionall in Shankes: And contrariwise: 4 and 5. p. vj.
The comparison both of the rate and proportion of triangles hath in the former beene taught: Their similitude remaineth for the last place. Which similitude of theirs consisteth indeed of the reason, or rate of their angles and proportion of the shankes. Therefore for just cause was the reason of the angles set first: Because from thence not onelyonly their reason, but also their latter proportion is gathered. Let a e i and i o u, be two triangles equiangled: And let them be set upon the same line e i u, meeting or touching one another in the common point i. Then, seeing that the angles at e and i, are granted to bee equall, the lines … i, and a e, are parallell, by the 21 e v. Therefore by the 22 e v u o and e a, being continued, shall meete. Item, The right lines a i … and y u, by the 21 e v, are parallell, because the angle a i e is equall to o u i, the inner opposite to it. Therefore seeing that a i is parallell to the base y u, by the 21 e v, e a shall be to a y, that is, by the 26 e v, to i …, as e i is to i u: And alternly, or crosse wayes, e a shall be to e i, as i o is to i u. This is the first proportion. Item, seeing that i o is parallell to the base y e; y o, that is, by the 26 … v, au shall bee unto o u, as e i, is unto i u: And crosse wise, as a i is unto i e, so is o u unto u i. This is the second proposition. Lastly, equiordinately: a e is to a i, as o i is to o u: wherefore if triangles be equiangled, they are proportional in shankes.
This converse is thus demonstrated. Let there be two triangles a e i, and o u y, proportionall in shankes … And as a e is to e i; so let o u, be to u y: And as a i is to … e; so let o y bee to y u. Then at the points u and y, let angles be made by the … … e iij. equall to the angles at e and i, and let the triangle u y s, be made: for the other angles at a and s, sha…l be equall by the 4 e. And the triangle y a …, shall be equiangled to the assigned a e i. And by the an…ecedent, it shall be proportionall to it in shankes. Thus are two triangles o u y, by the grant; and … y s, by the construction, proportionall in shanks to the same triangle a e i: And as a e, is to e i, so is e u, to u y; so is s u, to u y. Therefore seeing o u and s u, are proportionall to the same y u, they are equall; Item, as a i is to i e: so is o y unto y u: so also is s y unto y u. Therefore o y and s y, seeing they are proportionall to the same y u, are equall… (y u is the common side.) The triangle therefore o u y, is equilater unto the triangle s y u: And by the i e, it is to it equiangle: And therefore it is equiangled to the triangle a e i, which was to be prooved. This was generally before taught at the 20 e iiij, of homologall sides subtending equall angles.
Therefore,
13. If a right line in a triangle be parallell to the base, it doth cut off from it a triangle equiangle to the …hole… but lesse in base.
As in the triangle a e i, the right line o u, doth cut off the triangle a o u, equiangle, by the 21 e v, to the whole a e i; But the base o u, is lesse than the base e i, as appeareth by the 21 e, and by the alternation of the sides.
14. If two trangles be proportionall in the shankes of the equall angle, they are equiangles: 6 p vj.
Let therefore the triangles given be a e …, and o u y … equall in their angles a and oe And in their shankes let e a, be unto a i, as o u is to o y: And by the … e iij, let the angles s o y … and o y s, be equall to the angles e a i, and e i a: The other at s and e, shall be equall, by the 4 e. Here thou seest that the triangle a e i, is equiangle unto … y s. Now, by the 12 e. as e a is to a i: so is s o to o y: and therefore, by the grant, so is u o to o y. Therefore seeing that u o, and o s, are proportionall to o y, they are both equall. Lastly, if the common shanke o y bee added to both the shankes o u, and o y, are equall to the shankes s o and o y. [But by the construction the angles … … … and a i e are equall. And, by the 4 e, the other at s and e are equall Therefore the first triangle a e i, is made equiangled to the third. Now seeing the second triangle u o y is to the third s o y, equall in the shanks of the equall angle, it is to the same equilater, and by the i e, equiangled: Shon.] Wherefore the second triangle o u y shall likewise be equiangled to o s y, the third: And therefore if two triangles proportional in shankes be equall in the angle of their shankes, they are equiangles.
15 If triangles proportionall in shankes, and al…ernly parallell, doe make an angle betweene them, their bases are but one right line continued. 32 p. vj.
Or thus: If being proportionall in their feet, and alternately parallels, they make an angle in the midst betweene them, they have their bases continued in a right line: H.
The cause is out of the 14 e v. For they shall make on each side, with the falling line a i, two angles equall to two right angles.
Let the triangles a e i and o i u, be proportionall in shanks: As a e is to a i, so let i o be to o u: And let e a bee parallell to i o: And a i to ou: Item, let them make the angle a i o … betweene them, to wit, betweene their m… dle shankes a i, and o i, I say their bases e i, and i u, are bu… one right line continued. For seeing that by the grant a e, and o i, are parallels: Item a i and u o, the right line a i and o i, shall make, by the 21 e v, the angles at a, and o, equall to the alterne angle a i o: And therefore they are equall betweene themselves: And then, by the 14 e, the triangles given are equiangles: Therefore the angle o u i … is equall to the angle a i e: Wherfore the three angles o i u, o i a, and a i e, by the 3 e, are equall to the three angles of the triangle e a i, which are equall by the 13 e vj. VntoUnto two right angles: And therefore they themselves also are equall to two right angles. Wherefore, by the 14 e v, e i, and i u, are one right line continued.
16 If two triangles have one angle equall, another proportionall in shankes, the third homogeneall, they are equiangles. 7. p. v. j.
Let a e i, and o u y, the triangles given be equall in their angles a, and o: and proportionall in the shankes of the angles e, and u: and their other angles, at i, and y, homogeneall, that is, let them be both, either acute, or obtuse, or right angles. But first let them be acute, I say, the other at e, & u, are equall. Otherwise let a e s, by the 11 e iij. be made equall to the same o u y; Then have you them by the 4 e, equiangles; and the angles a s e, shall be equall to the angle o y u; and both are acute angles: and by the 12. e, a e s, and o u y, are proportionall in fides: and as a e, is to e s; so shall o …, be to u y, that is, by the grant, so shall a e, be to e i. Therefore because the same e a, hath unto two, to wit, e s, and e i, the same reason, the said e s, and e i, are equall one to another: And therefore, by the 17. e. v.j. the angles at the base in s and i, are equall. Therefore both of them are acute angles: And in like manner a s e, is an acute angle, contrary to the 14. e v. The same will fall out altogether like to both the other, being either obtuse or right angles. The last part of a right angle is manifest by the 4 e of this Booke.