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to AC (I. 12), and join EB, EC, ED; and because the straight line EF, which passes through the centre, cuts the straight line AC, which does not pass through the centre, at right angles, it shall likewise bisect it (III. 3); therefore AF B is equal to FC; and because the straight line AC is bisected in F, and produced to D, the rectangle AD DC, together with the square of FC, is equal to the square of FD; to each of these equals add the square of FE; therefore the rectangle AD DC, together with the squares of CF, FE, is equal to the squares. of DF, FE; but the square of ED is equal to the squares of DF, FE, because EFD is a right angle; and the square of EC is equal to the squares of CF, FE; therefore the rectangle AD DC, together with the square of EC, is equal to the square of ED; and CE is equal to EB; therefore the rectangle AD DC, together with the square of EB, is equal to the square of ED; but the squares of EB, BD, are equal to the square of ED, because EBD is a right angle; therefore the rectangle AD DC, together with the square of EB, is equal to the squares of EB, BD; take away the common square of EB; therefore the remaining rectangle ADDC, is equal to the square of DB.

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COR.-If from any point without a circle there be drawn two straight lines cutting it, as AB, AC, the rectangles contained by the whole lines and the parts of them without the circle, are equal to one another; namely, the rectangle BAAE, to the rectangle CA AF; for each of them is equal to the square of the straight line AD which touches the circle.

Schol. 1.—The above corollary and proposition 35, may be enunciated thus:-The rectangles under the segments of two intersecting chords of a circle are equal, whether they cut internally or externally.

Schol. 2. The first case of this proposition affords a geometrical principle by which the length of the diameter of the earth may be computed.

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If from a point without a circle a secant be drawn, and also a line meeting the circle, and if the rectangle under the secant and its external segment be equal to the square of the other line, this line will be a tangent.

Let any point D be taken without the circle ABC, and from it let two straight lines DCA and DB be drawn, of which DCA cuts the circle, and DB meets it; if the rectangle AD DC, be equal to the square of DB, DB touches the circle.

Draw the straight line DE touching the circle ABC (III. 17); find the centre F, and join FE, FB, FD; then FED is a right angle (III. 18); and because

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DE touches the circle ABC, and DCA cuts it, the rectangle AD DC, is equal to the square of DE (III. 36); but the rectangle AD DC, is, by hypothesis, equal to the square of DB; therefore the square of DE is equal to the square of DB; and the straight line DE equal to the straight line DB; and FE is equal to FB, wherefore DE, EF, are equal to DB, BF; and the base FD is common to the two triangles DEF, DBF; therefore the angle DEF is equal to the angle DBF (I. 8); and DEF is a right angle; therefore also DBF is a right angle; but FB, if produced, is a diameter, and the straight line which is drawn at right angles to a diameter, from the extremity of it, touches the circle (III. 16); therefore DB touches the circle ABC.

EXERCISES.

1. A line that bisects two parallel chords in a circle, is also perpendicular to them.

2. Parallel chords in a circle intercept equal arcs.

3. The exterior angle of a quadrilateral figure inscribed in a circle, is equal to the interior and opposite.

4. If two circles cut each other, the line joining the points of intersection is bisected perpendicularly by the line joining their centres.

5. If a tangent to a circle be parallel to a chord, the point of contact bisects the intercepted arc.

6. Two concentric circles intercept between them, two equal portions of a line cutting them both.

7. If a chord to the greater of two concentric circles be a tangent to the less, it is bisected in the point of contact. 8. If a circle be described on the radius of another circle, any chord in the latter, drawn from the point in which the circles meet, is bisected by the former.

9. If two circles touch one another, either internally or externally, the chords of the two arcs intercepted by two lines drawn through the point of contact, are parallel.

10. If two circles cut one another, and, from one of the points of intersection, two diameters be drawn, their other extremities, and the other point of contact, will be in one straight line.

11. If two chords in a circle intersect each other perpendicularly, the sum of the squares of their four segments is equal to the square of the diameter.

12. Perpendiculars, from the extremities of a diameter of a circle, upon any chord, cut off equal segments.

13. In a circle, the sum of the squares of two lines drawn from the extremities of a chord, to any point in a diameter parallel to it, is equal to the sum of the squares of the ments of the diameter.

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14. Given the vertical angle, the base, and the altitude of a triangle, to construct it.

15. Given the vertical angle, the base, and the sum of the sides of a triangle, to construct it.

16. If the opposite sides of a quadrilateral inscribed in a circle be produced to meet, the square of the line joining the points of concourse, is equal to the sum of the squares of the two tangents from these points.

17. If the points of contact of two tangents to a circle be joined, any secant drawn from their intersection is divided into three segments such, that the rectangle under the secant and its middle segment is equal to that under its extreme segments.

18. Two tangents to a circle, drawn from the same point, are equal.

FOURTH BOOK.

DEFINITIONS.

1. A rectilineal figure is said to be inscribed in another rectilineal figure, when all the angles of the inscribed figure are upon the sides of the figure in which it is in

scribed, each upon each.

2. In like manner, a figure is said to be described about another figure, when all the sides

of the circumscribed figure pass through the angular points of the figure about which it is described, each through each. 3. A rectilineal figure is said to be inscribed in a circle, when all the angles of the inscribed figure are upon the circumference of the circle.

4. A rectilineal figure is said to be described about a circle when each side of the circumscribed figure touches the circumference of the circle.

5. In like manner, a circle is said to be inscribed in a rectilineal figure, when the circumference of the circle touches each side of the figure.

6. A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure about which it is described.

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7. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle. 8. A regular polygon has all its sides equal, and also all its angles.

9. A polygon of five sides is called a pentagon; of six, a hexagon; of seven, a heptagon; of eight, an octagon; of nine, a nonagon; of ten, à decagon; of eleven, an undecagon; of twelve, a dodecagon; and of fifteen, a quindecagon or pentedecagon.

10. The centre of a regular polygon is a point equally distant from its sides or angular points.

11. The apothem of a regular polygon is a perpendicular from its centre upon any of its sides.

12. The perimeter of any figure is its circumference or whole boundary.

PROPOSITION I. PROBLEM.

In a given circle to place a straight line, equal to a given straight line not greater than the diameter of the circle.

Let ABC be the given circle, and D the given straight line, not greater than the diameter of the circle.

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Draw BC a diameter of the circle ABC; then, if BC is equal to D, the thing required is done; for in the circle ABC a straight line BC is placed equal to D; but if it is not, BC is greater than D; make CE equal to D (I. 3), and from the centre C, at the distance CE, describe the circle AEF, and join CA; therefore, because C is the centre of the circle AEF, CA is equal to CE; but D is equal to CE; therefore D is equal to CA. Wherefore, in the circle ABC, a straight line is placed, equal to the given straight line D, which is not greater than the diameter of the circle.

PROPOSITION II. PROBLEM.

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In a given circle to inscribe a triangle equiangular to a given triangle.

Let ABC be the given circle, and DEF the given triangle; it is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF.

Draw the straight line GAH touching the circle in the point A (III. 17), and at the point A, in the straight line AH, make the angle HAC equal to

the angle DEF (I. 23); and at the point A, in the straight line AG, make the angle GAB equal to the angle DFE, and join BC; therefore, E because HAG touches the circle

ABC, and AC is drawn from the point of contact, the angle

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