The Elements

[...of the] lines are all equal to each other; but [whether] the lines joined from the points of bisection? to the subsequent angles are broken or not, they were not shown.

12

If the diameter of a circle, or a diameter [intersecting] any of the chords in the circle, is extended in a straight line to any point outside; and from the point, a diameter is drawn where then? if it meets a certain point? of the circle's circumference, the lines joined [are] toward the opposite circumference [and] a certain point [are] on the same seated? angles, and toward the sides joined from the diameter—let them happen to be in the middle of the ab line segment ab diameter—certain diameters from the point to the circumference; the circle's circumference, the exi?, is opposite? if the ad and at point Theta, and they contain the angle phi with E; I say that the angle [under the] diameter... surrounding? ab from the diameter, the segment of the angle being understood as seated? on itself in the diagram, arrives at a certain vertex? of [that which is] in the through the same in the rest in on the angle is each...

A geometric diagram at the bottom of the page shows a circle. A horizontal diameter is drawn, and below it, an inscribed triangle or set of lines meets at a point on the lower circumference.

...if from the diameter [rising] on a certain area on a certain [part] on the equal [part], and from each point [lines] led through the center will make angles equal to the angles in the circle, as in the matter of the angles themselves, [in] the center, it is called a katheton perpendicular because of their standing upon the rest... toward the ab from every circumference, if [it is] of the segment from the katheton perpendicular of the ad they were shown. —

A geometric diagram in the right margin consists of a vertical line, a horizontal line extending to the right, and a slanted line originating from the vertex.

Of Ioannes, through toward the ab angles, the circumferences, the two toward all equal of the [things] of two, and from them out of the angle of the [things] and surrounding ab on—I say that it [is] itself for to it angles from two likewise the two which are considered equal [from] the point of the circle at ab to the given angle, it was shown in the circle from the point from the projection of the ab, if the [lines] through from the [points] into a straight line onto the ab, so that it [is] to the [other], if possible from angles in all [things] common, they fall upon it eb, but the [angles] under all best, the ba ab were shown some...

A geometric diagram at the bottom right shows an angle with its vertex at the bottom. A vertical ray and a slanted ray extend upwards. A horizontal line segment is positioned above the vertex, appearing to represent a plane or chord from which the lines originate.