61. PROP. XIII. Two parallel chords in any circle will intercept equal arcs.

Let AB, CD be any two parallel chords in the same circle; the arc AC shall be equal

to the arc BD.

Find E the centre of the circle; and draw EF perpendicular to AB, meeting the circumference of the circle in F.

Then since CD is parallel to AB, EF is also perpendicular to CD (34 Cor. 3); .. both the chords AB, CD are bisected by

A

F

B

the straight line EF (49 Cor.): and .. both the arcs AFB, CFD, are bisected in F (60); that is, arc AF= arc BF, and arc CF = arc DF; but if equals be taken from equals the remainders will be equal, .. arc AC=arc BD.

Conversely, if arc_AC=arc BD, the chord AB is parallel to the chord CD.

62. PROP. XIV. If the distance between the centres of two circles, which are in the same plane, be equal to the sum or difference of their radii, the circles will touch each other at one point only; and the point of contact will be in the straight line which joins the centres, or in that line produced.

1. Let A, and B be the centres of two circles so situated, in the same plane, that AB, the straight line joining the centres is equal to the sum of their radii.

Let C be the point in which AB meets the circumference of the

first circle, then AC=

A

B

D

the radius of that circle; and since AC + BC= the sum of the radii, BC must be the radius of the other circle; and C is a point in its circumference; that is, the two circumferences have the point C common to both. And they have no other point common: for, if CD be

drawn from C at right angles to AB, since CD is a tangent to both circles at the point C, every point in it, except the point C, is without both, that is, no point but is common to the two, and they touch each other in that point.

2. Let AB, the straight line joining the centres of the two circles, be equal to the difference of the radii. Produce AB to meet the circumference of the greater circle, whose centre is A, in C; then since AC is the radius of the greater circle, and AB is the difference of the two radii, BC= the radius of the smaller, and .. C is a point in the circum

ference of the latter; that is, C is a point common to both circumferences. That Cis the only point common to the two circumferences is shewn precisely as in the former case; and.. the circles touch each other at that point.

In the former case the circles are said to touch each other externally, in the latter internally.

63. PROP. XV. If a straight line touch a circle, and from the point of contact a chord be drawn dividing the circle into two segments, the angle between the tangent and this chord shall be equal to the angle in the alternate segment** of the circle.

E

Let the straight line ABC touch the circle BDE in the point B; and let BD be a chord dividing the circle into two segments. From B draw BE at right angles to AB, to meet the circumference again in E, which will .. be a diameter of the circle (55 Cor. 3). Join DE; take any point F in the arc BD, and join BF, DF. Then LCBD=LBED 'in the alternate segment'; and ABD = LBFD.

L

A

B

By alternate segment is meant the segment on the other side of the chord.

For, since BE is a diameter, BDE is a right angle (54), being the angle 'in a semicircle'; .. BED + ¿EBD = a right angle (37) = ≤ CBD + ¿EBD, .. ▲ CBD = ‹ BED.

Again, since BFDE is a 'quadrilateral inscribed in a circle', < BFD + 4 BED = two right angles (53) = 4 ABD +≤ CBD; and 4 CBD has been shewn to be equal to < BED, .. ‹ ABD = BFD.

[It might appear, at first sight, that by drawing BE at right angles to AB, we have proved only a particular case of the proposition; but it is not so, because BED=every other angle in the same segment' (52 Cor).]

EXERCISES B.

(1) Are all diameters of the same circle equal to one another? Shew that the diameter is greater than any other straight line drawn in the circle and terminated by the circumference.

(2) Does the chord of an arc increase as the arc increases? State the limitations.

(3) Can a circle be made up of segments? If so, of how many?

(4) Can a circle be made up of sectors? If so, of how many? In what case will a sector become a segment?

(5) Shew that the circumferences of circles which have the same centre cannot cut each other.

(6) If the circumference of a circle be divided into four equal arcs, shew that the chords of any two of them, which are adjacent, are at right angles to each other.

(7) If the circumference of a circle be divided into six equal parts, shew that the chord of each of them is equal to the radius.

(8) If the radius of a given circle be equal to a given straight line, find the centre of the circle.

(9) Make a circle of given radius, whose circumference shall pass through, 1st, one given point, 2ndly, two given points.

(10) Can more than one circle be drawn whose circumference shall pass through three given points?

(11) Shew that in particular cases a circle may be

drawn with its circumference passing through four, or a greater number of, given points. Exhibit such a case.

(12) In a given circle draw a chord which shall be both equal and parallel to a given chord in the same circle.

(13) If an arc or a segment of a circle be given, complete the circle.

(14) Through a given point within a given circle draw the least chord.

(15) Through a given point within a given circle draw a chord which shall be equal to a given line not greater than the diameter of the circle.

(16) If one circle intersect another, shew that the straight line joining the points of intersection is at right angles to the straight line joining their centres.

(17) Shew that the two tangents, which can be drawn to a circle from a point without it, are equal to one another.

(18) Shew that the straight line drawn through the middle point of an arc parallel to the chord of the arc is a tangent to the circle at that point.

(19) Can two distinct straight lines touch a circle at the same point?

(20) Divide a given arc into four equal parts.

(21) Have equal circles equal circumferences or perimeters? Is this the case with equal squares, triangles, and other rectilineal equal plane figures?

(22) If AB, CD be any two chords in a circle at right angles to each other, prove that the sum of the arcs AC, BD is equal to half the circumference.

(23) From two given points draw two straight lines which shall meet in a given straight line, and be at right angles to each other. Within what limits only is this possible?

(24) Apply Prop. vi. to draw a straight line at right angles to a given straight line from one extremity of it, when the given line cannot be produced'.

(25) Construct a square, when the diagonal only is given.

(26) If tangents be drawn to a circle from the extremities of a diameter, shew that any line intercepted between them, and touching the circle, subtends at the centre a right angle.

(27) Shew that about a given circle a certain number of equal circles can be drawn touching it and each other; and find the number.

(28) If two circles touch each other internally, and the radius of one of them be half that of the other, shew that every straight line, drawn from the point of contact to meet the outer circumference, is bisected by the inner one.

(29) A straight line touches a circle, and from the point of contact A any chord AB is drawn ; BC is another chord parallel to the tangent, and BD a chord parallel to AC. Shew that the chords AB, AC, CD are equal to one another.

(30) If the circumferences of two circles intersect each other, and through one of the points of intersection the diameters be drawn, shew that the other extremities of those diameters and the other point of intersection will be in one and the same straight line.

(31) Three equal circles are given, in the same plane, of which no two intersect each other, find the point from which if tangents be drawn to each circle, those tangents shall be equal to one another.

(32) What is the angle which the arc of a quadrant subtends at any point in the remaining portion of the circumference? Is it the same for all circles?

(33) In any two circles which have the same centre if a chord be drawn to the outer one and intersecting the inner one, shew that the parts of this chord intercepted between the two circumferences are always equal.

(34) If two circles touch each other, either externally or internally, and through the point of contact two straight lines be drawn forming four chords, two in each circle, shew that the straight lines joining the extremities of these chords in each circle are parallel to one another.

Such circles are sometimes called 'concentric' circles.