If a straight line drawn through the centre of a circle bisects a chord which does not pass through the centre, it is perpendicular to it; or, if perpendicular to it, it bisects it.

Two chords in a circle, intersecting in a point which is not the centre, cannot bisect each other.

If two circles intersect, they have not the same

centre.

E

If two circles touch internally, they have not the

If from a point within a circle, but which is not its centre, lines are drawn to the circumference; the greatest of those lines is that which passes through the centre, and the least is the remaining part of the diameter. Of the others, that which is nearer to the line passing through the centre, is greater than that which is more remote; the two lines which make equal angles with that passing through the centre, on opposite sides of it, are equal to each other; and there cannot be drawn a third line equal to them, from the same point to the circumference.

PROP. VIII. THEOR.

If from a point without a circle, straight lines are drawn to the circumference; of those falling upon the concave circumference the greatest is that which passes through the centre, and the line which is nearer the greatest is greater than that which is more remote: of those falling on the convex circumference the least is that which being produced would pass through the centre, and the line which is nearer to the least is less than that which is more remote; and lines, whether falling on the concave or convex circumference, which make equal angles with that passing through the centre, are equal to each other; and no third line can be drawn equal to them from the same point to the circumference.

PROP. IX. THEOR.

In a circle, a point from which more than two equal straight lines can be drawn to the circumference, must be the centre of the circle.

The circumferences of circles which intersect each other, cannot have more than two points in common.

If two circles touch each other internally, the straight line joining their centres will, when produced, pass through a point of contact.

PROP. XII. THEOR.

If two circles touch each other externally, the right line joining their centres will pass through a point of contact.

Two circles cannot touch each other either internally or externally in more than one point.

Equal straight lines inscribed in a circle are equally distant from the centre; and also, straight lines equally distant from the centre are equal.