Euclid's Elements

Book III



Book III    Propositions

Proposition 1.
To find the center of a given circle.

Proposition 2.
If on the circumference of a circle two points be taken at random, the straight line joining the points will fall within the circle.

Proposition 3.
If in a circle a straight line through the center bisect a straight line not through the center, it also cuts it at right angles; and if it cut it at right angles, it also bisects it.

Proposition 4.
If in a circle two straight lines cut one another which are not through the center, they do not bisect one another.

Proposition 5.
If two circles cut one another, they will not have the same center.

Proposition 6.
If two circles touch one another, they will not have the same center.

Proposition 7.
If on the diameter of a circle a point be taken which is not the center of the circle, and from the point straight lines fall upon the circle, that will be greatest on which the center is, the remainder of the same diameter will be least, and of the rest the nearer to the straight line through the center is always greater than the more remote, and only two equal straight lines will fall from the point on the circle, one on each side of the least straight line.

Proposition 8.
If a point be taken outside a circle and from the point straight lines be drawn through to the circle, one of which is through the center and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the center is the greatest, while of the rest the nearer to that through the center is always greater than the more remote, but, of the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote, and only two equal straight lines will fall on the circle from the point, one on each side of the least.

Proposition 9.
If a point be taken within a circle, and more than two equal straight lines fall from the point on the circle, the point taken is the center of the circle.

Proposition 10.
A circle does not cut a circle at more points than two.

Proposition 11.
If two circles touch one another internally, and their centers be taken, the straight line joining their centers, if it be produced, will fall on the point of contact of the circles.

Proposition 12.
If two circles touch one another externally, the straight line joining their centers will pass through the point of contact.

Proposition 13.
A circle does not touch a circle at more points than one, whether it touch it internally or externally.

Proposition 14.
In a circle equal straight lines are equally distant from the center, and those which are equally distant from the center are equal to one another.

Proposition 15.
Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the center is always greater than the more remote.

Proposition 16.
The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further the angle of the semi-circle is greater, and the remaining angle less, than any acute rectilineal angle.

Proposition 17.
From a given point to draw a straight line touching a given circle.

Proposition 18.
If a straight line touch a circle, and a straight line be joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent.

Proposition 19.
If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the tangent, the center of the circle will be on the straight line so drawn.

Proposition 20.
In a circle the angle at the center is double of the angle at the circumference, when the angles have the same circumference as base.

Proposition 21.
In a circle the angles in the same segment are equal to one another.

Proposition 22.
The opposite angles of quadrilaterals in circles are equal to two right angles.

Proposition 23.
On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.

Proposition 24.
Similar segments of circles on equal straight lines are equal to one another.

Proposition 25.
Given a segment of a circle, to describe the complete circle of which it is a segment.

Proposition 26.
In equal circles equal angles stand on equal circumferences, whether they stand at the centers or at the circumferences.

Proposition 27.
In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centers or at the circumferences.

Proposition 28.
In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to the less.

Proposition 29.
In equal circles equal circumferences are subtended by equal straight lines.

Proposition 30.
To bisect a given circumference.

Proposition 31.
In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; and further the angle of the greater segment is greater than a right angle, and the angle of the less segment less than a right angle.

Proposition 32.
If a straight line touch a circle, and from the point of contact there be drawn across, in the circle, a straight line cutting the circle, the angles which it makes with the tangent will be equal to the angles in the alternate segments of the circle.

Proposition 33.
On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilineal angle.

Proposition 34.
From a given circle to cut off a segment admitting an angle equal to a given rectilineal angle.

Proposition 35.
If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

Proposition 36.
If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of them cut the circle and the other touch it, the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference will be equal to the square on the tangent.

Proposition 37.
If a point be taken outside a circle and from the point there fall on the circle two straight lines, if one of them cut the circle, and the other fall on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference be equal to the square on the straight line which falls on the circle, the straight line which falls on it will touch the circle.

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