This is a consectary drawne out of the [28 e iiij]. For if the inscripts were halfed, they should be diameters, against the grant.
But rate hath beene hitherto in the parts of inscripts: Proportion in the same parts followeth.
12 If two inscripts doe cut one another, the rectangle of the segments of the one is equall to the rectangle of the segments of the other. 35 p iij.
If the inscripts thus cut be diameters, the proportion is manifest, as in the first figure. For the Rectangle of the segments, of the one is equall to the rectangle of the segments of the other, seeing they be both quadrates of equall sides. If they be not diameters let them otherwise as ae, and io: I say the Oblong of au, and ue, is equall to the Oblong of ou, and ui. For let the raies from the Center y, be ye, and yi. To the quadrate of each of these both the rectangles of the segments shall be equall. For by the [7 e], let the diameter yu, fall upon the point of the common section u; And let ys, and sr, be perpendiculars. Here by the [5 e xj]. the inscripts are cut equally in the points r and s: And unequally in the point u: Therefore by the [7 e xiij], the
oblong, of ou, and ui, with the quadrate su, is equall to the quadrates si; And adding ys, the same oblong, with the quadrates us and sy, that is, by the [9 e xij], with the quadrate yu, is equall to the quadrates is and sy, that is, by the [9 e xij], to the quadrate iy, that is, by the [5 e xij], to ye, to the which by the same cause it is manifest the other oblong with the quadrate yu is equall. Let the quadrate yu, bee taken from each of them: And then the oblongs shall be equall to the same: And therefore betweene themselves.
And this is the comparison of the parts inscripts. The rate of whole inscripts doth follow, the which whole one diameter doth make:
13 Inscripts are equall distant from the center, unto which the perpendiculars from the center are equall 4 d iij.
As it appeareth in the next figure, of the lines ae and io, unto which the perpendiculars uy and us, from the Center u, are equall.