Of Geometry the Eight Booke, of the Diverse Kindes of Triangles
1. A triangle is either right angled, or obliquangled.
The division of a triangle, taken from the angles, out of their common differences, I meane, doth now follow. But here first a speciall division, and that of great moment, as hereafter shall be in quadrangles and prismes.
2. A right angled triangle is that which hath one right angle: An obliquangled is that which hath none. 27. d j.
A right angled triangle in Geometry is of speciall use and force; and of the best Mathematicians it is called Magister matheseos, the master of the Mathematickes.
Therefore
3. If two perpendicular lines be knit together, they shall make a right angled triangle.
As here in aei. This construction and manner of making of a right angled triangle, is drawne out of the definition of a right angle. For right lines perpendicular are the makers of a right angle, as is manifest by the 13. e iij.
4. If the angle of a triangle at the base, be a right angle, a perpendicular from the toppe shall be the other shanke: [and contrariwise Schon.]
As is manifest in the same example.
5. If a right angled triangle be equicrurall, each of the angles at the base is the halfe of a right angle: And contrariwise.
As in the triangle aei: For they are both equall to one right angle, by the 13. e. vj. And betweene themselves, by the 17. e. vj.
Therefore
6. If one angle of a triangle be equall to the other two, it is a right angle [And contrariwise Schon.]
Because it is equall to the halfe of two right angles, by the 13. e vj.
And
7. If a right line from the toppe of a triangle cutting the base into two equall parts be equall to the bisegment, or halfe of the base, the angle at the toppe is a right angle: [And contrariwise Schon.]
As in the triangle aei, the right line ao, cutting the base ei, in o, into two equall parts, is equall to eo, or oi, the halfe of the base maketh two equicrural triangles; and the severall angles at the top equall to the angles at the ends, viz. e, and i, by the 17. e. vj. Therefore the angle at the toppe is equall to the other two: wherefore by the 6 e, it is a right angle.
8. A perpendicular in a triangle from the right angle to the base, doth cut it into two triangles, like unto the whole and betweene themselves, 8. p vj. [And contrariwise Schon.]
As in the triangle aei, the perpendicular ao, doth cut the triangles aoe, and aoi, like unto the whole aei, because they are equiangles to it; seeing that the right angle on each side is one, and another common in i, and e: Therefore the other is equall to the remainder, by 4. e vij. Wherefore the particular triangles are equiangles to the whole: As proportionall in the shankes of the equall angles, by the 12. e vij. But that they are like betweene themselves it is manifest by the 22. e iiij.
Therefore
9. The perpendicular is the meane proportionall betweene the segments or portions of the base.
As in the said example, as io, is to oa: so is oa, to oe, because the shankes of equall angles are proportionall, by the 8. e. From hence was Platoes Mesographus invented.
And
10. Either of the shankes is proportionall betweene the base, and the segment of the base next adjoyning.
For as ei, is unto ia, in the whole triangle, so is ai, to io, in the greater. For so they are homologall sides, which doe subtend equall angles, by the 23. e. iiij. Item, as ie, is to ea; in the whole triangle, so is ae, to eo, in the lesser triangle.
Either of the shankes is proportionall betweene the summe, and the difference of the base and the other shanke. And contrariwise. If one side be proportionall betweene the summe and the difference of the others, the triangle given is a rectangle. M. H. Brigges.
This is a consectary arising likewise out of the 4 e. of very great use.
In the triangle ead, the shanke ad, 12. is the meane proportionall betweene bd, 18. (the summe of the base ae, 13. and the shanke ed, 5.) and 8. the difference of the said base and shanke: For if thou shalt draw the right lines ba, and ac, the angle bac, shall be by the 6. e, a rectangle; (because it is equall to the angles at b, and c, seeing that the triangles bea, and eac, are equicrurall.) And by the 9 e, bd, da, and dc, are continually proportionall.
If a quadrate of a number, given for the first shanke, be divided of another, the halfe of the difference of the divisour, and quotient shall be the other shanke, and the halfe of the summe shall be the base. Or thus, The side of divided number doubled, and the difference of the divisour and quotient, shall be the two shankes, and the summe of them shall be the base.
Let the number given for the first shanke be 4. And let 8. divide 16. the quadrate of 4. by 2. The halfe of 8 - 2, that is 3. shall be the other shanke: And the halfe of 8 + 2, that is 5. shall be the base.
Therefore
If any one number shall divide the quadrate of another, the side of the divided, and the halfe of the difference of the divisour and the quotient, shall be the two shankes of a rectangled triangle, and the halfe of the summe of them shall be the base thereof.
Let the two numbers given be 4. and 6. The square of 6. let it be 36. and the quotient of 36. by 4. be 9: And the side is 6. for the one shanke. Now 9 - 4. that is, 5. is the difference of the divisour and quotient, whose halfe 2.½, is the other shanke. And 9 + 4. that is 13. is the summe the said devisour and quotient, whose halfe 6.½, is the base.
Againe let 4. and 8. be given. The quadrate of 8. is 64. And the quotient of 64 is 16. and the side of 64. is 8. for the one shanke. The halfe 16 - 4. that is 6. is the other shanke. And the halfe of 16 + 4. that is 10, is the base.
11. If the base of a triangle doe subtend a right-angle, the rectilineall fitted to it, shall be equall to the like rectilinealls in like manner fitted to the shankes thereof: And contrariwise, out of the 31. p. vj.
Or thus: If the base of a triangle doe subtend a right angle, the right lined figure made upon the base, is equall to the right lined figures like, and in like manner situate upon the feete: H.
Let the right angled triangle be aei: and let there be also the triangles eau, and aiy, and to them upon the base of the said right angle, by the 23 e iiij. let the triangle ies, be made like, and in like manner situate. I say, that eis, is equall joyntly to eau, and aiy. Let ao, a perpendicular fall from the right angle a, to the base ei: This by the ioe, doth yeeld us twise three proportionals, to wit, ie, ea, eo: Item, ei, ia, io: Therefore, by the 25. e. iiij, as ie, is to eo: so is the triangle ies, to the triangle eau; And as ei, is to oi, so is the triangle eis, to the triangle aiy: But ei, is equall to eo, and oi, the whole, to wit, to his parts. Wherefore by the second composition in Arithmeticke (9. c. ij.) the triangle eis, is equall to the triangles eau, and iay.
The Converse is thus proved: Let the triangle be aei: And let the perpendicular eo, be erected upon ae, equall to ei: And draw a right line from o to a: Here by the former, the rectilinealls situate at oe, and ea, that is by the construction, at ae, and ie, are equall to the rightilineall at ao, made alike and situate alike: And by the graunt they are equall, to the rectilineall at ai, made alike and situated alike. Therefore seeing the like rectilineals at ao, and ai, are equall; they have by the 20 e iiij, their homologall sides equall: And the two triangles are equiliters: And by the 1 e vij, equiangles. But aeo, is a right angle, by the construction: And aei, is proved to be equall to the same aeo: Therefore, by the 13 e v. aei, also is a right angle.
12. An obliquangled triangle is either Obtusangled or Acutangled.
The division of an obliquangled triangle is taken from the speciall differences of an oblique angle. For at the 15 e iij, we were taught that an oblique angle was either obtuse or acute: Therefore an obliquangled triangle is an obtuseangle, and an Acutangle.
13. An obtusangle is that triangle which hath one blunt corner. 28.d i.
There can be but one right angle in a triangle, by the 2 e. Therefore also in it there can be but one blunt angle.
Therefore
14. If the obtuse or blunt angle be at the base of the triangle given, a perpendicular drawne from the toppe of the triangle, shall fall without the figure: And contrarywise.
As here in aei, the perpendicular io, falleth without: This is manifest by the 4 e.
And
15. If one angle of a triangle be greater than both the other two, it is an obtuse angle: And contrariwise.
This is plaine by the 6 e.
And
16. If a right line drawne from the toppe of the triangle cutting the base into two equall parts, be lesse than one of those halfes, the angle at the toppe is a blunt-angle. And contrariwise.
As in aei, the perpendicular eo, cutting the base ai into two equall parts ao, and oi: And the said eo is lesse than either ao, or oi: Therefore the angle aei, is a blunt angle by the 7 e.
17. An acutangled triangle is that which hath all the angles acute. 29 d j.
Therefore
18. A perpendicular drawne from the top falleth within the figure: And contrariwise.
As in aei, the perpendicular ao falleth within as is plaine by the 4 e.
And
19. If any one angle of triangle be lesse then the other two, it is acute: And contrariwise.
As is manifest by the 6 e.
And
20. If a right line drawne from the toppe of the triangle; cutting the base into two equall parts, be greater than either of those portions, the angle at the toppe is an acute angle: And contrariwise.
As in aei, let ao cutting the base ei into two equall parts, be greater than any one of those parts, the angle at the toppe is an acuteangle, as appeareth by the 7 e.