The Way to Geometry, Book 17
Of Geometry the seventeenth Booke, Of the Adscription of a Circle and Triangle.
HItherto we have spoken of the Geometry of Rectilineall plaines, and of a circle: Now followeth the Adscription of both: This was generally defined in the first book 12 e. Now the periphery of a circle is the bound therof. Therefore a rectilineall is inscribed into a circle, when the periphery doth touch the angles of it 3 d iiij. It is circumscribed when it is touched of every side by the periphery; 4 d iij.
1. If a rectilineall ascribed unto a circle be an equilater, it is equiangle.
Of the inscript it is manifest; And that of a Triangle by it selfe: Because if it be equilater, it is equiangle, by the 19 e vj. But in a Triangulate the matter is to be prooved by demonstration. As here, if the inscripts o u, and s y, be equall, then doe they subtend equall peripheries, by the 32 e x v. Then if you doe omit the periphery in the middest betweene them hoth, as here u y, and shalt adde o i e s the remainder to each of them, the whole o i e s y, subtended to the angle at u: And u o i e s, subtended to the angle at y, shall be equall. Therefore the angles in the periphery, insisting upon equall peripheries are equall.
Of the circumscript it is likewise true, if the circumscript be understood to be a circle. For the perpendiculars from the center a, unto the sides of the circumscript, by the 9e xij, shal make triangles on each side equilaters, & equiangls, by drawing the semidiameters unto the corners, as in the same exāple.
2. It is equall to a triangle of equall base to the perimeter, but of heighth to the perpendicular from the center to the side.
As here is manifest, by the 8 e vij. For there are in one triangle, three triangles of equall heighth.
The same will fall out in a Triangulate, as here in a quadrate: For here shal be made foure triangles of equall height.
Lastly every equilater rectilineall ascribed to a circle, shall be equall to a triangle, of base equall to the perimeter of the adscript. Because the perimeter conteineth the bases of the triangles, into the which the rectilineall is resolved.
3. Like rectilinealls inscribed into circles, are one to another as the quadrates of their diameters, 1 p. x i j.
Because by the 1 e vj, like plains have a doubled reasō of their homologall sides. But in rectilineals inscribed the diameters are the homologall sides, or they are proportionall to their homologall sides. As let the like rectangled triangles be a e i, and o u y; Here because a e and o u, are the diameters, the matter appeareth to be plaine at the first sight. But in the Obliquangled triangles, s e i, and r u y, alike also, the diameters are proportionall to their homologall sides, to wit, e i and u y. For by the grant, as s e is to r u: so is e i to u y, And therefore, by the former, a s the diameter e a and u o.
In like Triangulates, seeing by the 4 e x, they may be resolved into like triangles, the same will fall out.
Therefore
4. If it be as the diameter of the circle is unto the side of rectilineall inscribed, so the diameter of the second circle be unto the side of the second rectilineall inscribed, and the severall triangles of the inscripts be alike and likely situate, the rectilinealls inscribed shall be alike and likely situate.
This Euclide did thus assume at the 2 p xij, and indeed as it seemeth out of the 18 p vj. Both which are conteined in the 23 e iiij. And therefore we also have assumed it.
Adscription of a Circle is with any triangle: But with a triangulate it is with that onelyonly which is ordinate: And indeed adscription of a Circle is common to all.
5. If two right lines doe cut into two equall parts two angles of an assigned rectilineall, the circle of the ray from their meeting perpendicular unto the side, shall be inscribed unto the assigned rectilineall. 4 and 8. p. iiij.
As in the Triangle a e i, let the right lines a o, and e u, halfe the angles a and e: And from y, their meeting, let the perpendiculars unto the sides be y o, y u, y s; I say that the center y, with the ray y o, or y a, or y s, is the circle inscribed, by the 17 e x v. Because the halfing lines with the perpendiculars shall make equilater triangles, by the 2 e vij. And therefore the three perpendiculars, which are the bases of the equilaters, shall be equall.
The same argument shall serve in a Triangulate.
6. If two right lines do right anglewise cut into two equall parts two sides of an assigned rectilineall, the circle of the ray from their meeting unto the angle, shall be circumscribed unto the assigned rectilineall. 5 p iiij.
As in the former figures. The demonstration is the same with the former. For the three rayes, by the 2 e vij, are equall: And the meeting of them, by the 17 ex, is the center.
And thus is the common adscription of a circle: The adscription of a rectilineall followeth, and first of a Triangle.
7. If two inscripts, from the touch point of a right line and a periphery, doe make two angles on each side equall to two angles of the triangle assigned be knit together, they shall inscribe a triangle into the circle given, equiangular to the triangle given è 2 p iiij.
Let the Triangle a e i be given: And the circle o, into which a Triangle equiangular to the triangle given, is to be inscribed. Therefore let the right line u y s, touch the periphery y r l: And from the touch y, let the inscripts y r, and y l, make with the tangent two angles u y r, and s y l, equall to the assigned angles a e i, and a i e: And let them be knit together with the right line r l: They shall by the 19 e xvj, make the angle of the alterne segments equall to the angles u r l, and y r l. Therefore by the 4 e vij seeing that two are equall, the other must needs be equall to the remainder.
The circumscription here is also speciall.
8 If two angles in the center of a circle given, be equall at a common ray to the outter angles of a triangle given, right lines touching a periphery in the shankes of the angles, shall circumscribe a triangle about the circle given like to the triangle given. 3 p iiij.
Let there be a Triangle, and in it the outter angles a e i, and a o u: The Circle let it be y s r; And in the center l, let the angles y l r, and s l r; at the common side l r, bee made equall to the said outter angles a e i, and a o u. I say the angles of the circumscribed triangle, are equall to the angles of the triangle given. For the foure inner angles of the quadrangle y l r m, are equall to the foure right angles, by the 6 e x: And two of them, to wit, at y and r, are right angles, by the construction: For they are made by the secant and touch line, from the touch point by the center, by the 20 e x v. Therefore the remainders at l and m, are equall to two right angles: To which two a e i and a e o are equall. But the angle at l, is equall to the outter: Therefore the remainder m, is equall to a e o. The same shall be sayd of the angles a o e, and a o u. Therefore two being equall, the rest at a and i, shall be equall.
Therefore
9. If a triangle be a rectangle, an obtusangle, an acute angle, the center of the circumscribed triangle is in the side, out of the sides, and within the sides: And contrariwise. 5 e iiij.
As, thou seest in these three figures underneath, the center a.