(59.) THEOREM. If, from the central point of any parallelogram, as a centre, a circle be described with any radius, the sum of the squares of the four lines drawn from any point in the circumference, to the four corners of the parallelogram, will always amount to the

same sum.

D

B

From G the centre of the parallelogram ABCD, let a circumference of a circle be described, having any radius; also from F any point in this circumference, let lines be drawn to the four corners of the parallelogram. Then, (Art. 58), twice the sum of the squares of these four lines will be equal to the sum of the squares of the diagonals, together with eight times the square of the radius FG; but the diagonals and the radius always remain the same, and therefore the squares of the four lines thus drawn will always amount to the same sum.

BOOK THIRD.

DEFINITIONS.

1. Any portion of the circumference of a circle is called

an arc.

2. The straight line joining the extremities of an arc is called a chord. The chord is said to subtend the arc.

3. The portion of the circle included by an arc and its chord, is called a segment.

H

Thus the space FGDF, included by the arc FGD and the chord FD, is a segment; so also is the space included by the same chord and the arc FAHBD.

4. The portion included between two radii and the intercepted arc, is called a sector.

The space BCH is a sector.

5. When a straight line touches the circumference in only one point, it is called a tangent; and the common point of the line and circumference is called the point of

contact.

6. One circle touches another, when their circumferences have only one point common.

7. A line is inscribed in a circle, when its extremities are in the circumference.

8. An angle is inscribed in a circle, when its sides are inscribed.

9. A polygon is inscribed in a circle when its sides are inscribed; and under the same circumstances, the circle is said to circumscribe the polygon.

Thus AB is an inscribed line, ABC an inscribed angle, and the figure ABCDF an inscribed poly

gon.

10. A circle is inscribed in a

polygon when its circumference

F

D

touches each side, and the polygon is said to be circumscribed about the circle.

11. By an angle in a segment of a circle, is to be understood an angle whose vertex is in the arc, and whose sides intercept the chord of said arc; and by an angle at the centre, is meant one whose vertex is at the centre. In both cases, the angles are said to be subtended by the chords or arcs which their sides include.

12. The circumference of a circle may be described by causing the extremity B of the line AB to revolve about the other extremity A, which remains fixed. In this revolution, while the line AB passes over the angular space BAC, the extremity B passes

D

C

B

A

over the arc BC; and while the line passes over the angular space CAD, its extremity describes the arc CD; and so for other angles. Hence the angles at the centre are measured by the arcs included between their sides.

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PROPOSITION I.

THEOREM. If a line, drawn through or from the centre of a circle, bisect a chord, it will be perpendicular to the chord; or, if the line be perpendicular to the chord, it will bisect both the chord and the arc of the chord.

Let AB be any chord in a circle, and CD a line drawn from the centre C to the chord; then, if the chord be bisected at the point D, CD will be perpendicular to AB.

A

D

F

Drawing the two radii CA, CB, and comparing the two triangles ACD, BCD, which have CA equal to CB (Def. XXIII), and CD common, as also AD equal to DB by hypothesis, we find that the three sides of the one are equal to the three sides of the other; therefore the angle CDA is equal to the angle CDB (B. I, Prop. vi), and each of these angles is a right angle, and CD is perpendicular to AB (Def. X). Again, if CD be perpendicular to AB, then will the chord AB be bisected at the point D, or have AD equal to DB; and the arc AFB will be bisected at the point F, or have AF equal to FB.

For, drawing CA, CB as before, we have the triangle CAB isosceles, and consequently the angle CAD is equal to CBD (B. I, Prop. v); we also have the angles CDA and CDB equal, each being a right angle (Def. X); therefore the third angles of these triangles are equal (B. I, Prop. XXIV, Cor. 1), and, having the side CD common,

G

they must also have the side AD equal to the side DB (B. I, Prop. IV).

Also, since the angle ACF is equal to the angle BCF, the arc AF which measures the former is equal to the arc BF which measures the latter.

Cor. Hence a line bisecting any chord at right angles, passes through the centre of the circle.

(60.) By means of this proposition, we may find the centre of a given circle, or of a given arc of a circle.

:

Let ABC be the arc of a circle it is required to find its centre.

Draw any two chords AB, BC; bisect them with the perpendiculars DF and GH; then will the point K, in which they intersect, be the centre sought.

For, by the corollary of the above pro

position, each of the perpendiculars DF and GH passes through the centre, and therefore the centre is at the point K.

(61.) It is also obvious that the above operation is equivalent to finding the centre of a circle whose circumference will pass through three given points A, B, C. Hence, by this means, a circumference of a circle may always be made to pass through any three points, not situated in the same straight line.

(62.) The same method is applicable, in case it is required to describe a circle about a triangle.

PROPOSITION II.

THEOREM. If two circles, the one placed within the other, touch, the centres of the circles and the point of contact will be in the same straight line.