104.

ELEMENTS OF

PROPOSITION C. THEOREM.

ED.

If from any point in the circumference of a circle a perpendicular be drawn to a diameter, and a chord to either end of it, the square of the chord is equal to the rectangle under the diameter and the part of it between the perpendicular and the chord.

From any point C in the circumference of the circle ACBD let a perp. CE and a chord CA be drawn to the diameter AB; AC2 = AB.AE. For AC2 = CEAE (B. 2), and AB. AE = AEAE. EB (3.2). But CEAE. EB (2 Cor.

35. 3); therefore AB. AF

+CE, or AC. Therefore, if

AE2

from any point &c. Q. E. D.

C

B

E

PROPOSITION D. THEOREM.

ED.

If two chords of a circle intersect each other, the angle which they contain is equal to half the angle at the centre which stands on the sum or the difference of the arches intercepted between them, according as they meet within or without the circle.

1. Let ACBD be a circle, and AB, CD two chords which intersect each other in the point E within the circle; the angle DEB is equal to half the angle at the centre which stands on an arch equal to the sum of the arches AC, DB.

=

=

=

Through C draw CG parallel to AB, then the arch BG AC (A.3), and therefore the arch DBG AC G + DB. Now the angle DEB angle DCG (29. 1), which is equal to half the angle at the centre standing on the same arch DBG (20. 3); therefore the angle DEB is equal

B

D

to half the angle at the centre standing on the arch DBG.

Cor. If two chords intersect each other at right angles within a circle, the sum of the opposite arches which they inter- . cept is equal to half the circumference.

PROPOSITION E. THEOREM.

ED.

If from any point without a circle a tangent and a secant be drawn, the angle which they contain is equal to an angle at the circumference of the circle standing on an arch equal to the difference between the arches intercepted by the tangent and secant.

From any point E without the circle BDC let a tangent EBA and a secant EDC be drawn; the angle AFC is equal to an angle at the circumference on an arch equal to the differ ence of the arches BC, BD.

From the point D draw DF parallel to AE, then the arch FB is equal to BD (A. 3), therefore the arch FC is equal to the difference of the arches BC and BD.

Again, because FD is parallel to AE, the angle FD is equal to AEC (29. 1). But FIC is an angle at the circumference standing on the arch FC, which is equal to the dif

A

E

D

B

C

ference of the arches BC, BD. Therefore, if from &c. Q.E.D.

P

THE PRINCIPAL THEOREMS IN BOOK III.

If in a circle a straight line bisect a chord at right angles, it will pass through the centre of the circle.

If in a circle a diameter bisect a chord, it will be perpendicular to the chord; and if it be perpendicular to a chord, it will bisect the chord, and also the arch subtended by the chord.

Chords in a circle, which are equal to one another, are equally distant from the centre; and chords which are equally distant from the centre of a circle are equal to one another.

A straight line perpendicular to the extremity of a radius or a diameter is a tangent to the circle.

If a straight line touch a circle, a radius drawn to the point of contact is perpendicular to the tangent.

An angle at the centre of a circle is double of an angle at the circumference, standing on the same arch. All angles in the same segment of a circle, or standing on the same arch, are equal to one another.

The two opposite angles of any quadrilateral figure described in a circle are together equal to two right angles.

In equal circles equal angles stand on equal arches, whether they be at the centres or at the circumferences of the circles.

In equal circles the angles which stand on equal arches are equal to one another, whether they be at the centres or at the circumferences of the circles.

In equal circles equal chords subtend equal arches on the same sides of them, and equal arches are subtended by equal chords.

An angle in a semicircle is a right angle.

If a straight line touch a circle, and a chord be drawn from the point of contact, the angle made by the tangent and chord is equal to the angle in the segment of the circle which is on the other side of the chord.

If two chords in a circle cut each other, the rectangle contained by the segments of one chord is equal to the rectangle contained by the segments of the other.

If from any point without a circle two secants be drawn, the rectangle contained by one of them and its external part is equal to the rectangle contained by the other and its external part.

If from any point without a circle a tangent and a secant be drawn, the rectangle contained by the secant and its external part is equal to the square of the tangent.

If from any point in the circumference of a circle a perpendicular be drawn to a diameter, and a chord to either end of it, the square of the perpendicular is equal to the rectangle under the segments of the diameter; and the square of the chord is equal to the rectangle of the diameter and its segment adjacent to the chord.

If two chords of a circle intersect each other, the angle which they contain is equal to half the angle at the centre which stands on the sum or the difference of the arches intercepted between them, according as they meet within or without the circle.

END OF BOOK III.