M. On the Relations of straight Lines drawn from any Point to the III. 7. . III. 7, schol. III. 9... III. 8.. Circumference of a Circle. HYPOTHESES. If from any point within a circle, which is not the center, straight lines be drawn to the circumfer ence. Idem. If a point be taken within a circle, from which more than two equal straight| lines can be drawn to the circumference. If from any point without a circle straight lines be drawn to the circumference. Of those which fall on the concave circumference, the greatest is that which passes through the center. Of the rest, that which is nearer to the line passing through the center is greater than the more remote. But of those which fall on the convex circumference, the least is that which, if produced, would pass through the center. Of the rest, that which is nearer to the least is less than the more remote. And more than two straight lines cannot be drawn, either to the concave or convex circumference, which shall be equal. Those lines which form equal angles with the line passing through the center equal. The rectangle under the whole line which cuts the circle and the segment without the circle, is equal in area to the square on the line which touches it. are HYPOTHESES. III. 36, cor. 1.. If from a point without a circle two straight lines be drawn, cutting it. If from a point without a circle two straight lines be drawn, one cutting the circle and the other meeting it, and if the rectangle under the whole line which cuts the circle, and the part of it without the circle, be equal in area to the square on the line which meets it. If from a point within or without a circle two straight lines be drawn at right angles to each other, to meet the circumference. If from any point in the diameter of a circle or its extensions straight lines be drawn to the end of a parallel chord. If in a circle two straight lines cut one another, which do not both pass through the center. If two straight lines cut one another within a circle. If a straight line drawn through the center of a circle bisect a straight line which does not passthrough the center, And if it is perpendicular to it. If from the middle point of a finite straight line as a center a circle be described, and lines be drawn from any point in its circumference to the extremities of the line. If a perpendicular be drawn from any point in the circumference of a semicircle to the diameter. CONSEQUENCES. -The rectangles under the whole lines, and the parts of them without the circle, are equal in area to one another. That line is a tangent to the circle. The sum of the squares on the segments between the point and the circumference is equal in area to the square on the diameter of the circle. The squares on those lines are together equal in area to the squares on the segments into which the point divides the diameter. They do not bisect each other. -The rectangle under the segments of one of them is equal in area to the rectangle under the segments of the other. >It is perpendicular to it. It bisects it. The sum of the squares on those lines is always the same, and equal in area to double the sum of the squares on the radius and half the given line. The square on the perpendicular is equal in area to the rectangle under the segments into which it divides the diameter. N. On the mutual Contact of a straight Line and a Circle. HYPOTHESES. If a straight line cut the circumference of a circle. If a straight line touches the circumference of a circle. Idem. If tangents are drawn from the same point to a circle. If tangents are at the extremities of the same diameter. If a straight line touches the circumference of a circle, and a straight line be drawn perpendicular to it from the point of contact. If a straight line be drawn from the extremity of the diameter of a circle, perpendicular to the same, And if any straight line be drawn from a point between that perpendicular and the circle, to the point of contact. If a circle be described on the radius of another circle, and a straight line be drawn from the point in which they meet to the outer circumference. CONSEQUENCES. It cannot do so in more than two points. It meets it in only one point. -The straight line drawn from the center to the point of contact shall be perpendidicular to the line touching the circle. They are equal to one another. They are parallel to one another. The center of the circle shall be in that line. >It will fall without the circle. It will cut the circumference of the circle. That line will be bisected by the interior one. If angles are in the same seg-They are equal to one an ment of a circle. If, in a circle, an angle be in a semicircle. But if the angle be in a seg ment greater than a semicircle. And if the angle be in a seg ment less than a semicircle. If an angle at the center of a circle have the same part of the circumference for its base as an angle at the circumference. If a straight line touches a circle, and from the point of contact a straight line be drawn cutting the circle. other. It is a right angle. It is less than a right angle. It is greater than a right angle. The former angle is double the latter. The angles formed by this line and the line touching the circle are equal to the angles in the alternate segments of the circle. III. 24. III. 14. III. 15. III. 26, cor. 1. . III. 26, cor. 2. III. 26, cor. 3. III. 26, cor. 4. III. 26, cor. 5. III. 28. III. 29. III. 32, cor. HYPOTHESES. If two segments of circles are similar, and upon equal straight lines. If two straight lines in a circle are equal, And if straight lines are equally distant from the center. If straight lines be drawn in a circle, of which one passes through its center. If two chords of a circle are parallel. If two chords of a circle meet one another. If chords intersect at the same angle, within a circle, And if they intersect without the circle, But if one pair intersect within, and the other without the circle. If two chords intersect within a circle at right angles. If a chord of a circle be produced till the produced part is equal to the radius, and if a line be drawn from its extremity through the center of the circle to meet the concave circumference. If, in equal circles, or the same circle, straight lines are equal. If in equal circles, or the same circle, equal parts of the circumference are taken. If two straight lines be drawn through the point of contact of two circles. CONSEQUENCES. They are equal to one another, and have equal arcs. They are equally distant from the center. They are equal to one another. That line is the greatest, And of all others, that which is nearer to the center is greater than the more re mote, And the greater is nearer to the center than the less. They intercept equal arcs. The angle formed by them is equal to the angle terminated at the circumference by the sum or difference of the arcs which they intercept, according as the point in which they meet is within or without the circle. The sums of the arcs which they respectively intercept are equal; The differences are equal; The sum of the one pair of arcs is equal to the difference of the other. The sums of the opposite arcs intersected are equal. The concave portion of the circumference intercepted is equal to three times the convex. They cut off equal parts of the circumferences, the greater equal to the greater, and the less to the less. They are subtended by equal straight lines. They intercept arcs the chords of which are parallel. |