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.