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.
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.
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.
From a given circle to cut off a segment admitting an angle equal to a given rectilineal angle.
On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilineal angle.
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.
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.
In equal circles equal circumferences are subtended by equal straight lines.
In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to the less.
In equal circles angles have the same ratio as the circumferences on which they stand, whether they stand at the centres or at the circumferences.
In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or at the circumferences.
In equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences.
Given a segment of a circle, to describe the complete circle of which it is a segment.
Similar segments of circles on equal straight lines are equal to one another.
On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.
The opposite angles of quadrilaterals in circles are equal to two right angles.
In a circle the angles in the same segment are equal to one another.
In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same circumference as base.
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 centre of the circle will be on the straight line so drawn.
If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight line so joined will be perpendicular to the tangent.
From a given point to draw a straight line touching a given circle.
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 semicircle is greater, and the remaining angle less, than any acute rectilineal angle.
Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the centre is always greater than the more remote.
To describe a circle with any centre and distance.
In a circle equal straight lines are equally distant from the centre, and those which are equally distant from the centre are equal to one another.
A circle does not touch a circle at more points than one, whether it touch it internally or externally.
In a given circle to inscribe a fifteen-angled figure which shall be both equilateral and equiangular.
If two circles touch one another externally, the straight line joining their centres will pass through the point of contact.
And the bases are the circles described by the two sides opposite to one another which are carried round.
In a given circle to inscribe an equilateral and equiangular hexagon.
If two circles touch one another internally, and their centres be taken, the straight line joining their centres, if it be also produced, will fall on the point of contact of the circles.
About a given pentagon, which is equilateral and equiangular, to circumscribe a circle.
A circle does not cut a circle at more points than two.
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 centre of the circle.
In a given pentagon, which is equilateral and equiangular, to inscribe a circle.
And the base is the circle described by the straight line which is carried round.
About a given circle to circumscribe an equilateral and equiangular pentagon.
- 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 centre and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the centre is greatest, while of the rest
the nearer to that through the centre 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.
- If on the diameter of a circle a point be taken which is not the centre of the circle, and from the point straight lines fall upon the circle, that will be greatest on which the centre is, the remainder of the same diameter will be least, and of the rest
the nearer to the straight line through the centre 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.
A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.
In a given circle to inscribe an equilateral and equiangular pentagon.
A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.
If two circles touch one another, they will not have the same centre.
About a given square to circumscribe a circle.
And the point is called the centre of the circle.
The centre of the sphere is the same as that of the semicircle.
If two circles cut one another, they will not have the same centre.
Given two circles about the same centre, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.
The axis of the sphere is the straight line which remains fixed and about which the semicircle is turned.
A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;
If in a circle two straight lines cut one another which are not through the centre, they do not bisect one another.
In a given square to inscribe a circle.
When, the diameter of a semicircle remaining fixed, the semicircle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere.
If in a circle a straight line through the centre bisect a straight line not through the centre, it also cuts it at right angles; and if it cut it at right angles, it also bisects it.
About a given circle to circumscribe a square.
If on the circumference of a circle two points be taken at random, the straight line joining the points will fall within the circle.
In a given circle to inscribe a square.
To find the centre of a given circle.
If an equilateral triangle be inscribed in a circle, the square on the side of the triangle is triple of the square on the radius of the circle.
About a given triangle to circumscribe a circle.
Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another.
In a given triangle to inscribe a circle.
If in a circle which has its diameter rational an equilateral pentagon be inscribed, the side of the pentagon is the irrational straight line called minor.
About a given circle to circumscribe a triangle equiangular with a given triangle.
A sector of a circle is the figure which, when an angle is constructed at the centre of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.
If an equilateral pentagon be inscribed in a circle, the square on the side of the pentagon is equal to the squares on the side of the hexagon and on that of the decagon inscribed in the same circle.
In a given circle to inscribe a triangle equiangular with a given triangle.
If the side of the hexagon and that of the decagon inscribed in the same circle be added together, the whole straight line has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon.
Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle.
A straight line is said to be fitted into a circle when its extremities are on the circumference of the circle.
An angle of a segment is that contained by a straight line and a circumference of a circle.
A segment of a circle is the figure contained by a straight line and a circumference of a circle.
A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed.
Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed.
A rectilineal figure is said to be circumscribed about a circle, when each side of the circumscribed figure touches the circumference of the circle.
In a circle straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal.
A rectilineal figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle.
Circles are said to touch one another which, meeting one another, do not cut one another.
Circles are to one another as the squares on the diameters.
A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.
Similar polygons inscribed in circles are to one another as the squares on the diameters.
Equal circles are those the diameters of which are equal, or the radii of which are equal.