PROP. XV. THEOR.

The diameter is the greatest of all straight lines inscribed in a circle; and of the others, that which is nearer to the centre is greater than that which is more remote; and, conversely, the greater line is nearer to the centre than the less.

A straight line drawn from the extremity of the diameter of a circle, and perpendicular to it, is a tangent to the circle: and through the same point there cannot be drawn another straight line also a tangent to the circle.

PROP. XVII. PROB.

To draw a tangent to a given circle from a given point, either in or outside of its circumference.

If a straight line touches a circle, the ray drawn to the point of contact is perpendicular to the tangent.

PROP. XIX. THEOR.

If a straight line touches a circle, the straight line drawn from the point of contact perpendicular to the tangent, passes through the centre of the circle.

The angle at the centre of a circle, is double of the angle at the circumference, standing on the same arch.

All angles in the same segment of a circle are

equal to one another.

PROP. XXII. THEOR.

The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles.

PROP. XXIII. THEOR.

Upon the same straight line, and upon the same side of it, there cannot be two similar segments of circles not coinciding with each other.

PROP. XXIV. THEOR.

Similar segments of circles upon equal straight lines are equal to each other.

A segment of a circle being given, to describe the circle of which it is a segment.

PROP. XXVI. THEOR.

In equal circles the arches on which stand equal angles, whether at the centre or circumference, are equal.