Of all the rectangles contained by the segments of a given straight line, the greatest is the square which is described on half the line.

PROP. XXVIII. PROB.

To divide a given straight line so that the rectangle contained by its segments may be equal to a given area, not exceeding the square of half the line.

E

B

PROP. XXIX. PROB.

To produce a given straight line, so that the rectangle contained by the segments between the extremities of the given line and the point to which it is produced, may be equal to a given area.

PROP. XXXI. THEOR.

In a right-angled triangle, the rectilinear figure described upon the side opposite to the right angle, is equal to the similar and similarly described figures upon the sides containing the right angle.

B

PROP. XXXII. THEOR.

If two triangles, having two sides of the one proportional to two sides of the other, be joined at one angle, so that their homologous sides be parallel to each other, the remaining sides shall be in a straight line.

PROP. XXXIII. THEOR.

In equal circles, angles, whether at the centre or circumference, are in the same ratio to one another as the arches on which they stand: so also are sectors.