As here thou seest: For the bisegments, or two equall portions thus cut are two triangles of equall heighth 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 aei: And let ao, cut the base ei, in o unequally: And let uy be parallell from u, the middest of the base, unto the sayd ao. I say that yo shall divide the triangle into two equall portions. For let au 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 ayu, and you, are equall by the 8 e; because they are of equall height, and upon the same base.
Take away ysu, the common triangle; And you shall leave asy, and osu, equall betweene themselves: The common right lined figure ysui, let it be added to both the sayd equall triangles: And then oyi, shall be equall to aui, the halfe part; And therefore aeoy, 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 Mr. 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 aei, and aou: And the angles oau, and eai, let them be equall: And as ua is to ae, so let ia be unto ao: I say that the triangles aou, and eai, are equall. For eo being knit together with a right line, uao is unto oae, as ua is unto ae, by the 7 e:
And ia, unto ao, by the grant, is as eai is unto eao. Therefore uao, and eai, are unto eao proportionall: And therefore they are equall one to another.