Subtendere, to draw or straine out something under another; and in this place it signifieth nothing else but to make a line or such like, the base of an angle, arch, or such like. And subtendi, is to become or made the base of an angle, arch, of a circle, or such like: As here, let ai, be a greater side than ae, I say the angle at e, shall be greater than that at i. For let there be cut off from ai, a portion equall to ae,; and let that be io: then the angle aei, equicrurall to the angle oie, shall be greater in base, by the grant. Therefore the angle shall be greater, by the [9 e iij].
The converse is manifest by the same figure: As let the angle aei, be greater than the angle aie. Therefore by the same, [9 e iij]. it is greater in base. For what is there spoken
of angles in generall, are here assumed specially of the angles in a triangle.
23. If a right line in a triangle, doe cut the angle in two equall parts, it shall cut the base according to the reason of the shankes; and contrariwise. 3. p vj.
The mingled proportion of the sides and angles doth now remaine to be handled in the last place.
Let the triangle be aei; and let the angle eai, be cut into two equall parts, by the right line ao: I say, as ea, is unto ai, so eo, is unto oi. For at the angle i, let the parallell iu, by the [24. e v]. be erected against ao; and continue or draw out ea, infinitly; and it shall by the [20. e v]. cut the same iu, in some place or other. Let it therefore cut it in u. Here, by the [28. e v]. as ea, is to au, so is eo, to oi. But au, is equall to ai, by the [17. e]. For the angle uia, is equall to the alterne angle oai, by the [21. e v]. And by the grant it is equall to oae, his equall: And by the [21. e v]. it is equall to the inner angle aui; and by that which is concluded it is equall to uia, his equall. Therefore by the [17. e], au, and ai, are equall. Therefore as ea, is unto ai, so is eo, unto oi.
The Converse likewise is demonstrated in the same figure. For as ea, is to ai; so is eo, to oi: And so is ea, to au, by the 12 e: therefore ai, and au, are equall, Item the angles eao, and oai, are equall to the angles at u, and i, by the [21. e v]. which are equall betweene themselves by the [17. e].