But the base ei, is the side of the equilater quinquangle. For if two right lines halfing both the angles of a triangle which is the double of the remainder, be knit together with a right line, both one to another, and with the angles, shall inscribe unto a circle an equilater triangle, whose one side shall be the base it selfe: As here seeing the angles eoa, eoi, uio, uia, iao, are equal in the periphery, the peripheries, by the [7 e, xvj]. subtending them are equall: And therefore, by the [32 e, xv]. the subtenses ae, ei, io, ou, ua, are also equall. Now of those five, one is ae. Therefore a right line proportionall cut, doth thus make the adscription of a quinquangle: And from thence againe is afforded a line proportionally cut.
6 If two right lines doe subtend on each side two angles of an inscript quinquangle, they are cut proportionally, and the greater segments are the sides of the said inscript è 8, p xiij.
As here, Let ai, and eu, subtending the angles on each side aei, and eau: I say, That they are proportionally cut in the point s: And the greater segments si, and su, are equall to ae, the side of the quinquangle. For here two triangles are equiangles: First aei, and uae, are equall by the grant, and by the [2 e, vij]. Therefore the angles aie, and aes, are equall. Then aei, and ase, are equall: Because the
angle at a, is common to both: Therefore the other is equall to the remainder, by the [4 e, 7]. Now, by the [12. e, vij.] as ia, is unto ae, that is, as by and by shall appeare, unto is: so is ea, unto as: Therefore, by the [1 e, xiiij]. ia, is cut proportionally in s. But the side ea, is equall to is: Because both of them is equall to the side ei, that by the grant, this by the [17. e, vj]. For the angles at the base, ise, and ies, are equall, as being indeed the doubles of the same. For ise, by the [16. e vj]. is equall to the two inner, which are equall to the angle at u, by the [17 e vj]. and by the former conclusion. Therefore it is the double of the angles aes: Whose double also is the angle uei, by the [7 e. xvj]. insisting indeede upon a double periphery.
And from hence the fabricke or construction of an ordinate quinquangle upon a right line given, is manifest.
Therefore
7 If a right line given, cut proportionall, be continued at each end with the greater segment, and sixe peripheries at the distance of the line given shall meete, two on each side from the ends of the line given and the continued, two others from their meetings, right lines drawne from their meetings, & the ends of the assigned shall make an ordinate quinquangle upon the assigned.
The example is thus.