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BOOK III.

OF THE CIRCLE, AND THE INVESTIGATION OF THEOREMS DEPENDENT ON ITS PROPERTIES.

DEFINITIONS.

1. * A Curved Line is one whose consecutive parts, however small, do not lie in the same direction.

2. A Circle is a plane figure bounded by one uniformly curved line, all of the points of which are at the same distance from a certain point within, called the center.

3. The Circumference of a cir

cle is the curved line that bounds it.

4. The Diameter of a circle is a line passing through the center, and terminating at both extremities in the circumference. Thus, in the figure, Cis the center of the circle, the curved line AGBD is the circumference, and AB is a diameter.

M

E

N

Q

G

F

C

A

BL

P

D

H

5. The Radius of a circle is a line extending from the center to any point in the circumference. Thus, CD is a radius of the circle.

6. An Arc of a circle is any portion of the circumference.

* The first six of the above definitions have been before given among the general definitions of Geometry, but it was deemed advisable to reinsert them here.

7. A Chord of a circle is the line connecting the extremities of an arc.

8. A Segment of a circle is the portion of the circle on either side of a chord.

Thus, in the last figure, EGF is an arc, and EF is a chord of the circle, and the spaces bounded by the chord EF, and the two arcs EGF and EDF, into which it divides the circumference, are segments.

9. A Tangent to a circle is a line which, meeting the circumference at any point, will not cut it on being produced. The point in which the tangent meets the circumference is called the point of tangency.

10. A Secant to a circle is a line which meets the circumference in two points, and lies a part within and a part without the circumference.

11. A Sector of a circle is a portion of the circle included between any two radii and their intercepted arc.

Thus, in the last figure, the line HL, which meets the circumference at the point D, but does not cut it, is a tangent, D being the point of tangency; and the line MN, which meets the circumference at the points P and Q, and lies a portion within and a portion without the circle, is a secant. The area bounded by the arc BD, and the two radii CB, CD, is a sector of the circle.

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F

E

all found in the circumference

of the circle; and conversely, the circle is then said to be

circumscribed about the polygon.

14. A Regular Polygon is one which is both equiangu

lar and equilateral.

The last three definitions are illustrated by the last figure.

THEOREM I.

Any radius perpendicular to a chord, bisects the chord, and

also the arc of the chord.

Let AB be a chord, C the center of the circle, and CE a radius perpendicular to AB; then we are to prove that AD = BD, and AE = EB.

C

D

A

B

E

Since C is the center of the circle, AC= BC, CD is common to the two A's ACD and BCD, and the angles at D are right angles; therefore the two A's ADC and BDC are equal, and AD = DB, which proves the first part of the theorem.

Now, as AD =DB, and DE is common to the two spaces, ADE and BDE, and the angles at D are right angles, if we conceive the sector CBE turned over and placed on CAE, CE retaining its position, the point B will fall on the point A, because AD = BD and AC = BC; then the arc BE will fall on the arc AE; otherwise there would be points in one or the other arc unequally distant from the center, which is impossible; therefore, the arc AE = the arc EB, which proves the second part of the theorem.

Hence the theorem.

Cor. The center of the circle, the middle point of the chord AB, and of the subtended arc AEB, are three points in the same straight line perpendicular to the chord at its middle point. Now as but one perpendicular can be drawn to a line from a given point in that line, it follows:

1st. That the radius drawn to the middle point of any arc bisects, and is perpendicular to, the chord of the arc.

2d. That the perpendicular to the chord at its middle point passes through the center of the circle and the middle of the subtended arc.

THEOREM II.

Equal angles at the center of a circle are subtended by

- equal chords.

Let the angle ACE = the angle

ECB; then the two isosceles triangles,
ACE, and ECB, are equal in all re-

C

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In the same circle, or in equal circles, equal chords are

equally distant from the center.

Let AB and EF be equal chords, and C the center of the circle. From C, draw CG and CH, perpendicular to the respective chords. These perpendiculars will bisect the chords, (Th. 1), and we shall have AG=EH. We are now to prove that CG = CH.

F

H

E

A

C

B

G

Since the A's ECH and ACG are right-angled, we

have, (Th. 39, В. І),

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AC2 (1)

EH - AG + HC2 - GC2 = EC But the chords are equal by hypothesis, hence their halves, EH and AG, are equal; also EC=AC, being radii of the circle. Wherefore,

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These values in Equation (1) reduce it to

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Hence the theorem.

Cor. Under all circumstances we have

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because the sum of the squares in either member of the equation is equivalent to the square of the radius of the circle.

Now, if we suppose HC greater than GC, then will HỠ2 be greater than GC2. Let the difference of these squares be represented by d.

2

Subtracting GC from both members of the above

equation, we have

whence,

EH2 + d = AG

AGEH2, and AG>EH.

Therefore, AB, the double of AG, is greater than EF, the double of EH; that is, of two chords in the same or

equal circles, the one nearer the center is the greater.

2

=

AG2 +

The equation, EH2 + HC2 GC, being true, whatever be the position of the chords, we may suppose GC to have any value between 0 and AC, the radius of the circle.

When GC becomes zero, the equation reduces to
EH2 + HC2 = AG2 = R^;

2

that is, under this supposition, AG coincides with AC, and AB becomes the diameter of the circle, the greatest chord that can be drawn in it.

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