Proposition 9: To describe a circle around an assigned quadrate.

Proposition 8: To describe a circle within an assigned quadrate.

Proposition 7: To describe a quadrate around a proposed circle.

Proposition 6: To describe a quadrate within a given circle.

Proposition 5: To describe a circle around an assigned triangle, whether it may be orthogonal, amblygonal, or oxygonal.

Proposition 4: To describe a circle within a given triangle.

Proposition 3: Around an assigned circle, to describe a triangle equiangular to an assigned triangle.

Proposition 2: Within an assigned circle, to assemble a triangle equiangular to an assigned triangle.

continuing from here: https://no-outlet.com/@ivlia/113749821408334309

Proposition 1: Within a given circle, to fit a right line equal to a given right line that is no greater than the diameter.

Proposition 36: If a point is marked without a circle whence two lines are drawn to the circumference, one cutting and the other applied to the circumference, and that made from the whole of the secant drawn according to the extrinsic part of it is equal to that made from the applied line drawn according to itself, from necessity the applied line will be touching the circle.

& that's book III. which was honestly pretty good. comparatively.

IV contd: https://no-outlet.com/@ivlia/113964625370391243

Proposition 35: If a point is marked without a circle whence two straight lines are drawn to the circle, with one line cutting and the other touching, then that contained within the whole secant as well the extrinsic part of it, is equal to the quadrate that is drawn from the tangent line.

Proposition 34: If in a circle two straight lines divide one another, that which proceeds within the two parts of one of them is equal to the rectangle that is contained within the two parts of the other line.

Proposition 33: From a given circle, to abscind a portion taking an angle equal to a given angle.

Proposition 32: Over a given line, to describe a portion of a circle taking an angle equal to a given angle, be it either right or greater or less than right.

Proposition 31: If a straight line contacts a circle and from the point of contact an other straight line is drawn within the circle, dividing the circle off center, whatever two angles it makes at the tangent are equal to the two angles that are over the arc in alternate portions of the circle.

Proposition 30: If a rectilinear angle in a semicircle rests upon the arc, it is right. And if it's in a portion less than a semicircle, it's greater than right. And if it's in a portion greater than a semicircle, it's less than right. Furthermore, the angle of any portion greater than a semicircle will be greater than right and by necessity, that of a lesser portion will be less than right.

Proposition 29: To divide a given arc by equals.

Proposition 28: It is necessary for equal arcs of equal circles to have equal chords.

Proposition 27: If equal lines resect arcs within equal circles, the arcs too shall be equal. And if the lines are not equal then the arcs too shall not be equal, and it is necessary for a greater arc to be abscinded by a greater line and a lesser arc by a lesser line.

Proposition 26: If equal arcs are assumed on equal circles, it is necessary for the angles fashioned beneath them, which are placed either on their centers or on their circumferences, to be equal.