Euclid. # A text-book of Euclid's Elements : for the use of schools : Books I-VI and XI online

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C" in A and B ; and let P'AQ' be the

limiting position of PQ when the point

B is brought into coincidence with A :

then shall CA be perp. to P'Q'.

Bisect AB at E and join CE:

then CE is perp. to PQ. iii. B.

Now let the secant PABQ change

its position in such a way that while the

point A remains fixed, the point B con-

tinually approaches A, and ultimately

coincides with it ;

then, Jiowever near B approaches to A, the st. line CE is always

perp. to PQ, since it joins the centre to the middle point of the chord

AB.

But in the limiting position, when B coincides with A, and the

secant PQ becomes the tangent P'Q', it is clear that the point E will

also coincide with A; and the perpendicular CE becomes the radius

CA. Hence CA is perp, to the tangent P'Q' at its point of contact

A. * Q.E.D.

NoTK. It follows from Proposition 2 that a straight line cannot

cut the circumference of a circle at more than two points. Now when

the two points in which a secant cuts a circle move towards coinci-

dence, the secant ultimately becomes a tangent to the circle: we

infer therefore that a tangent cannot meet a circle otherwise than

at its point of contact. Thus Euclid's definition of a tangent may be

deduced from that given by the Method of Limits.

214

KLCLIDm t'.I.K.MKM>.

2. By this Method Proposition 32 may he derived us a special case

from rropositio)i 21.

For let A and B be two points on the O"

of the Â©ABC;

and let BCA, BPA be any two angles in

the segment BCPA :

then the z BPA^^^tho z BCA. in. 21.

Produce PA to Q.

Now let the point P continually approach

the fixed point A, and ultimately coincide

with it ;

then, however near P may approach to A,

the Z BPQ=rthe z BCA. iii. 21.

But in the limiting position when

P coincides with A,

and the secant PAQ becomes the tangent AQ',

it is clear that BP will coincide with BA,

and the Z BPQ becomes the z BAQ'.

Hence the z BAQ' = the z BCA, in the alternate segment. q. k. d.

The contact of circles may be treated in a similar manner by

adopting the following definition.

Definition. If one or other of two intersecting circles alters its

position in such a way that the two points of intersection continually

approach one another, and ultimately coincide ; in the limiting posi-

tion they are said to toucli one another, and the point in which the

two points of intersection ultimately coincide is called the point of

contact.

EXAMPLES ON LIMITS.

1. Deduce Proposition 19 from the Corollary of Proposition 1

and Proposition 3.

2. Deduce Propositions 11 and 12 from Ex. 1, page 156.

3. Deduce Proposition G from Proposition 5.

4. Deduce Proposition 13 from Proposition 10.

5. Shew that a straight line cuts a circle in two different points,

TWO coincident points, or not at all, according as its distance from the

centre is less than, equal to, or greater than a radiiis.

6. Deduce Proposition 32 from Ex. 3, page 188.

7. Deduce Proposition 36 from Ex. 7, page 209.

8. The -angle in a semi-circle is a rifjht angle.

To what Theorem is this statement reduced, when the vertex of

the right angle is brought into coincidence with an extremity of the

diameter?

9. From Ex. 1, page 190, deduce the corresponding property of a

triangle inscribed in a circle.

THEOREMS AND EXAMPLES ON BOOK III. 215

THEOREMS AND EXAMPLES ON BOOK III.

I. ON THE CENTRE AND CHORDS OF A CIRCLE.

See Propositions 1, 3, 14, 15, 25.

1. All circles tvhich j^ass through a fixed point, and have then-

centres on a (jlven straight line, pass also throxigh a second fixed point.

Let AB be the given st. line, and P the given point.

P'

From P draw PR perp. to AB ;

and produce PR to P', making RP' equal to PR.

Then all circles which pass through P, and have their centres on

AB, shall pass also through P'.

For let C be the centre of amj one of these circles.

Join CP, CP'.

Then in the a" CRP, CRP'

{CR is common,

and RPâ€” RP', Constr.

and the z CRP = the z CRP', being rt. angles ;

.-. CP = CP'; 1.4.

.'. the circle whose centre is C, and which passes through P, must

pass also through P'.

But C is the centre of any circle of the system ;

.*. all circles, which pass through P, and have their centres in AB,

pass also through P'. q. e. d.

2. Describe a circle that shall pass through three given points not

in the same straight line.

216 KUCLID'h Kr,KMKNTS.

3. Describe a circle that shall pass through two given points and

have its centre in a given straight line. When is this impossible?

4. Describe a circle of given radius to pass through two given

points. When is this impossible?

5. ABC is an isosceles triangle ; and from the vertex A, as centre,

a circle is described cutting the base, or the base produced, at X and Y.

Shew that BX = CY.

6. If two circles which intersect are cut by a straight line

parallel to the common chord, shew that the parts of it intercepted

between the circumferences are equal.

7. If two circles cut one another, any two straight lines drawn

through a point of section, making equal angles with the common

chord, and terminated by the circumferences, are equal. [Ex. 12,

p. 156.]

8. If two circles cut one another, of all straight lines drawn

through a point of section and terminated by the circumferences, the

greatest is that which is parallel to the line joining the centres.

9. Two circles, whose centres are C and D, intersect at A, B;

and through A a straight line PAQ is drawn terminated by the

circumferences: if PC, QD intersect at X, shew that the angle PXQ

is equal to the angle CAD.

10. Through a point of section of two circles which cut one

another draw a straight line terminated by the circumferences and

bisected at the point of section.

11. AB is a fixed diameter of a circle, whose centre is C; and

from P, any point on the circumference, PQ is drawn perpendicular

to AB; shew that the bisector of the angle CPQ always intersects the

circle in one or other of two fixed points.

12. Circles are described on the sides of a quadrilateral as

diameters: shew that the common chord of any two consecutive

circles is parallel to the common chord of the other two. [Ex. 9,

p. 97.]

13. Two equal circles touch one another externally, and through

the point of contact two chords are drawn, one in each circle, at

right angles to each other : shew that the straight line joining their

other extremities is equal to the diameter of either circle.

14. Straight lines are drawn from a given external point to the

circumference of a circle : find the locus of their middle points.

[Ex. 3, p. 97.]

15. Two equal segments of circles are described on opposite sides

of the same chord AB ; and through O, the middle point of AB, any

straight line POQ is drawn, intersecting the arcs of the segments at

P and Q : shew that O P - OQ.

theor?:ms and examples on book iir. 217

II. ON THE TANGENT AND THE CONTACT OP CIRCLES.

See Propositions 11, 12, 16, 17, 18, 19.

1. All equal chords placed in a given circle touch a fixed concen-

tric circle.

2. If from an external point two tangents are drawn to a circle,

the angle contained by them is double the angle contained by the

chord of contact and the diameter drawn through one of the points of

contact.

3. Two circles touch one another externally, and through the

point of contact a straight line is drawn terminated by the circun>

ferences : shew that the tangents at its extremities are parallel.

4. Two circles intersect, and through one point of section any

straight line is drawn terminated by the circumferences : shew that

the angle between the tangents at its extremities is equal to the angle

between the tangents at the point of section.

5. Shew that two parallel tangents to a circle intercept on any

third tangent a segment which subtends a right angle at the centre.

6. Two tangents are drawn to a given circle from a fixed external

point A, and any third tangent cuts them produced at P and Q: shew

that PQ subtends a constant angle at the centre of the circle.

7. In any quadrilateral circumscribed about a circle, the suvi of

one pair of opposite sides is equal to the sum of the other pair.

8. If the sum of one pair of opposite sides of a quadrilateral is

equal to the sum of the other pair, shew that a circle may be inscribed

in the figure.

[Bisect two adjacent angles of the figure, and so describe a circle to

touch three of its sides. Then prove indirectly by means of the last

exercise that this circle must also touch the fourth side.]

9. Two circles touch one another internally: shew that of all

chords of the outer circle which touch the inner, the greatest is that

which is perpendicular to the straight line joining the centres.

10. In a right-angled triangle, if a circle is described from the

middle point of the hypotenuse as centre and with a radius equal to

half the sum of the sides containing the right angle, it will touch

the circles described on these sides as diameters.

11. Through a given point, draw a straight line to cut a circle,

so that the part intercepted by the circumference may be equal to a

given straight line.

In order that the problem may 6e possible, between what limits

must the given line lie, when the given point is (i) without the circle,

(ii) within it?

218 kuclid'b klkmknts.

12. A Belies of circles touch a given straight line at a given point:

shew that the tangents to them at the points where they cut a given

parallel straight line all touch a fixed circle, whose centre is the given

point.

13. If two circles touch one another internally, and any third

circle be described touching both ; then the sum of the distances of

the centre of this third circle from the centres of the two given circles

is constant..

14. Find the locus of points such that the pairs of tangents

drawn from them to a given circle contain a constant angle.

15. Find a point such that the tangents drawn from it to two

given circles may be equal to two given straight lines. When is this

impossible?

16. If three circles touch one another two and two ; prove that

the tangents drawn to them at the three points of contact are con.

current and equal.

The Common Tangents to Two Circles.

17. 'To draw a common tangent to two circles.

First, if the given circles are external to one another, or if they

intersect.

Let A be the centre of the

greater circle, and B the centre

of the less.

From A, with radius equal

to the diff^** of the radii of the

given circles, describe a circle:

and from B draw BC to touch

the last di'awn circle. Join AC,

and produce it to meet the

greater of the given circles at D.

Through B draw the radius BE pai^ to AD, and in the same

direction.

Join DE:

then DE shall be a common tangent to the two given circles.

For since AC = the diff'Â« between AD and BE, Comtr.

.'. CD=BE:

and CD is par^ to BE; Constr.

.-. DE is equal and par' to CB. i. 33.

But since BC is a tangent to the circle at C,

.-. the z ACB is a rt. angle; in. 18,

hence each of the angles at D and E is a rt. angle: i. 29.

.â€¢. DE is a tangent to both circles. q.e.f.

THEOREMS AND EXAMPLES ON BOOK III. 219

It follows from hypothesis that the point B is outside the circle

used in the construction :

.-. two tangents such as BC may always be drawn to it from B ;

hence two common tangents may always be drawn to the given

circles by the above method. These are called the direct common

tangents.

When the given circles are external to one another and do not

intersect, two more common tangents may be drawn.

For, from centre A, with a radius equal to the siuii of the radii of

the given circles, describe a circle.

From B draw a tangent to this circle ;

and proceed as before, but draw BE in the direction opposite to AD.

It follows from hypothesis that B is external to the circle used in

the construction ;

.-. two tangents may be drawn to it from B.

Hence tioo more conmion tangents may be drawn to the given

circles : these will be found to pass between the given circles, and are

called the transverse common tangents.

Thus, in general, four common tangents may be drawn to two

given circles.

The student should investigate for himself the number of common

tangents which may be drawn in the following special cases, noting

in each case where the general construction fails, or is modified : â€”

(i) When the given circles intersect :

(ii) When the given circles have external contact :

(iii) When the given circles have internal contact :

(iv) When one of the given circles is wholly within the other.

18. Drmo the direct common tangents to two equal circles.

19. If the two direct, or the two transverse, common tangents

are drawn to two circles, the parts of the tangents intercepted be-

tween the points of contact are equal.

20. If four common tangents are drawn to two circles external to

one another; shew that the two direct, and also the two transverse,

tangents intersect on the straight line which joins the centres of the

circles.

21. Two given circles have external contact at A, and a direct

common tangent is drawn to touch them at P and Q. : shew that PQ

subtends a right angle at the point A.

22. Two circles have external contact at A, and a direct common

tangent is drawn to touch them at P and Q: shew that a circle

described on PQ as diameter is touched at A by the straight line

which joins the centres of the circles.

220 EDCLTD's KliEMENTS.

23. Two circles whose centres are C and C have external contact

at A, and a direct common tangent is drawn to touch them at P

and Q : shew that the bisectors of the angles PCA, QC'A meet at

right angles in PQ. And if R is the point of intersection of the

bisectors, shew that RA is also a common tangent to tlie circles.

24. Two circles have external contact at A, and a direct common

tangent is drawn to touch them at P and Q : shew that the square

on PQ is equal to the rectangle contained by the diameters of the

circles.

25. Draw a tangent to a given circle, so that the part of it

intercepted by another given circle may be equal to a given straight

line. When is this impossible?

26. Draw a secant to two given circles, so that the parts of it

intercepted by the circumferences may be equal to two given straight

lines.

Problems on Tangency.

The following exercises are solved by the Method of Inter-

section of Loci, explained on page 117.

The student should begin by making himself familiar with

the following loci.

(i) The loais of the centres of circles which pass through two given

points.

(ii) The locus of the centres of circles xohich touch a given straight

line at a given point.

(iii) The locus of the centres of circles tchich touch a given circle at

a given point.

(iv) The locus of the centres of circles ivhich touch a given straight

line, and have a given radius.

(v) 'The locus of the centres of circles tchich touch a given circle,

and have a given radius.

(vi) 'The locus of the centres of circles which touch two given

straight lines.

In each exercise the student should investigate the limits

and relations among the data, in order that the problem may be

possible.

27. Describe a circle to touch three given straight lines.

28. Describe a circle to pass through a given point and touch a

given straight line at a given point.

29. Describe a circle to pass through a given point, and touch a

given circle at a given point.

THEOREMS AND EXAMPLES ON BOOK 111. 22 i

30. Describe a circle of given radius to pass through a given

point, and touch a given straight line.

31. Describe a circle of given radius to touch two given circles.

32. Describe a circle of given radius to touch two given straight

lines.

33. Describe a circle of given radius to touch a given circle and a

given straight line.

34. Describe two circles of given radii to touch one another and

a given straight line, on the same side of it.

35. If a circle touches a given circle and a given straight line,

shew that the points of contact and an extremity of the diameter of

the given circle at right angles to the given line are collinear.

36. To describe a circle to touch a given circle, and aho to touch a

given straight line at a given point.

Let DEB be the given circle, PQ.

the given st. line, and A the given

point in it :

it is required to describe a circle to

touch the Â© DEB, and also to touch

PQ at A.

At A draw AF perp. to PQ : 1. 11.

then the centre of the required circle

must lie in AF. iii. 19.

Find C, the centre of the Â© DEB,

I. 1.

and draw a diameter BD perp. to

PQ:

join A to one extremity D, cutting

the o"'' at E.

Join CE, and produce it to cut AF at F.

Then F is the centre, and FA the radius of the required circle.

[Supply the proof : and shew that a second solution is obtained by

joining AB, and producing it to meet the O"" :

also distinguish between the nature of the contact of the circles, when

PQ cuts, touches, or is without the given circle.]

37. Describe a circle to touch a given straight line, and to touch

a given circle at a given point.

38. Describe a circle to touch a given circle, have its centre in a

given straight line, and pass through a given point in that straight

line.

[For other problems of the same class see page 235.]

222 EUCLID'S KLEMENTS.

Oktuogonal Chicles.

Definition. Circles which intersect .'it a- point, so that the

two tangents at that point are at right angles to one another,

are said to be orthogonal, or to cut one another ortho-

gonally.

39. In two intersectiug circles the angle between the ta*)gents

at one point of intersection is equal to the angle between the tangents

at the other.

40. 7/ two circles cut one another orlJiononully, the tangent io

each circle at a point of intersection will p(/,sÂ« throwjh the centre of

the other circle.

41. If tico circles cut one another orthogonally, the nquare on the

distance between their centres is equal to the sum of the squares on

their radii.

42. Find the locus of the centres of all circles which cut a given

circle orthogonally at a given point.

43. Describe a circle to pass through a given point and cut a

given circle orthogonally at a given point.

III. ON ANGLES IN SEGMENTS, AND ANGLES AT THE

CENTRES AND CIRCUMFERENCES OF CIRCLES.

See Propositions 20, 21, 22; 26, 27, 28, 29; 31, 32, 33, 34.

1. If two chords intersect to i thin a circle, they form an angle equal

to that at the centre, subtended by half the sum of the arcs they cut off.

Let AB and CD be two chords, intersecting

at E within the given Â©ADBC:

then shall the l A EC be equal to the angle at

the centre, subtended by half the sum of the

arcs AC, BD.

Join AD.

Then the ext. z A EC = the sum of the int.

opp. /"EDA, EAD;

that is, the sum of the z ' CDA, BAD.

But the z-'CDA, BAD are the angles at

the O'^'' subtended by the arcs AC, BD ;

.-. their sum = half the sum of the angles at the centre subtended by

the same arcs;

or, the z A EC = the angle at the centre subtended by half the sum of

the arcs AC, BD. q. e. d.

THEOREMS AND EXAMPLES ON BOOK III. 223

2. If tivo chords lohen produced intersect outside a circle, they form

an angle equal to that at the centre subtended by half the difference of

the arcs they cut of.

B. The sum of the arcs cut off by two chords of a circle at right

angles to one another is equal to the semi-circumference.

4. AB, AC are any two chords of a circle; and P, Q are the

middle points of the minor arcs cut off by them: if PGl is joined,

cutting AB and AC at X, Y, shew that AX = AY.

5. If one side of a quadrilateral inscribed in a circle is inoduced,

the exterior angle is equal to the opposite interior angle.

6. If two circles intersect, and any straight lines are drawn, one

through each point of section, terminated by the circumferences;

shew that the chords which join their extremities towards the same

parts are parallel.

7. ABCD is a quadrilateral inscribed in a circle ; and the opposite

sides AB, DC are produced to meet at P, and CB, DA to meet at Q:

if the circles circumscribed about the triangles PBC, QAB intersect

at R, shew that the points P, R, Q are collinear.

8. If a circle is described on one of the sides of a right-angled

triangle, then the tangent drawn to it at the point where it cuts the

hypotenuse bisects the other side.

9. Given three points not in the same straight line : shew how

to find any uumber of points on the circle which passes through them,

without finding the centre.

10. Through any one of three given points not in the same

straight line, draw a tangent to the circle which passes through them,

without finding the centre.

11. Of two circles which intersect at A and B, the circumference

of one passes through the centre of the other : from A any straight

line ACD is drawn to cut them both; shew that CB = CD.

12. Two tangents AP, AQ are drawn to a circle, and B is the

middle point of the arc PQ, convex to A. Shew that PB bisects the

angle APQ.

13. Two circles intersect at A and B; and at A tangents are

drawn, one to each circle, to meet the circumferences at C and D : if

CB, BD are joined, shew that the triangles ABC, DBA are equi-

angular to one another.

14. Two segments of circles are described on the same chord and

on the same side of it ; the extremities of the common chord are joined

to any point on the arc of the exterior segment : shew that the arc

intercepted on the interior segment is constant.

224 Euclid's elements.

15. If a series of triangles are drawn standing on u lixed basi',

and having a given vertical angle, shew that the bisectors of the verti-

cal angles all pass through a fixed point.

16. ABC is a triangle inscribed in a circle, and E the middle

point of the arc subtended by BC on the side remote from A: ii"

through E a diameter ED is drawn, shew that the angle DEA is half

the difference of the angles at B and C. [See Ex. 7, p. 101.]

17. If two circles touch each other internally at a point A, any

chord of the exterior circle which touches the interior is divided at its

point of contact into segments which subtend equal angles at A.

18. If two circles touch one another internally, and a straight

line is drawn to cut them, the segments of it intercepted between the

circumferences subtend equal angles at the point of contact.

The Orthocen'tre of a Triangle.

I'J. The perpendiculars draini from the vertices of a Iriatiyle to

the opposite sides are concurrent.

In the A ABC, let AD, BE be the

perp' drawn from A and B to the oppo-

site sides ; and let them intersect at O,

Join CO; and produce it to meet AB

at F.

It is required to shew that CF is perp.

to AB.

Join DE.

Then, because the z ' OEC, GDC are

rt. angles, Hijp.

:. the points O, E, C, D are concyclic :

.-. the z DEC = the Z DOC, in the same segment;

= the vert. opp. Z FOA.

Again, because the z " AEB, ADB are rt. angles, Hyp.

:. the points A, E, D, B are concyclic:

.-. the Z DEB = the Z DAB, in the same segment.

.-. the sum of the Z " FOA, FAO = the sum of the z ' DEC, DEB

= a rt. angle : ^VP-

:. the remaining Z AFO = art. angle: i. 32,

that is, CF is perp. to AB.

Hence the three perp' AD, BE, CF meet at the point O. q. e. d.

[For an Alternative Proof see page lOG.]

THEOKKMS AND EXAMPLES ON BOOK III. 225

Definitions.

(i) The intersection of the perpendiculars drawn from the

vortices of a triangle to the opposite sides is called its ortho-

centre.

(ii) The triangle formed by joining the feet of the perpendi-

culars is called the pedal or orthocentric triangle.

20. In an acute-angled triangle the perpendiculars draivn from

the vertices to the opposite sides bisect the angles of the pedal triangle

through ivhich they pass.

In the acute-angled a ABC, let AD,

BE, CF be the perj)^ drawn from the

vertices to the opposite sides, meeting

at the orthocentre O; and let DEF be

the pedal triangle :

then shall AD, BE, CF bisect respect-

ively the z Â« FDE, DEF, EFD.

For, as in the last theorem, it may

be shewn that the points O, D, C, E are

concyclic ;

.â€¢. the Z ODE = the Z OCE, in the same segment.

Similarly the points O, D, B, F are concyclic;

,â€¢. the z ODF â€” the z OBF, in the same segment.

But the z OCE = the z OBF, each being the comp' of the z BAC.

.-. the Z ODE = the z ODF.

Similarly it may be shewn that the z " DEF, EFD are bisected by

BE and CF. q. e.d.

Corollary, (i) Every two sides of the pedal triangle are equally

inclined to that side of the original triangle in icliich they meet.

For the z EDC = the comp* of the Z ODE

= the comp' of the Z OCE

= the Z BAC.

Similarly it may be shewn that the z FDB = the Z BAC,

limiting position of PQ when the point

B is brought into coincidence with A :

then shall CA be perp. to P'Q'.

Bisect AB at E and join CE:

then CE is perp. to PQ. iii. B.

Now let the secant PABQ change

its position in such a way that while the

point A remains fixed, the point B con-

tinually approaches A, and ultimately

coincides with it ;

then, Jiowever near B approaches to A, the st. line CE is always

perp. to PQ, since it joins the centre to the middle point of the chord

AB.

But in the limiting position, when B coincides with A, and the

secant PQ becomes the tangent P'Q', it is clear that the point E will

also coincide with A; and the perpendicular CE becomes the radius

CA. Hence CA is perp, to the tangent P'Q' at its point of contact

A. * Q.E.D.

NoTK. It follows from Proposition 2 that a straight line cannot

cut the circumference of a circle at more than two points. Now when

the two points in which a secant cuts a circle move towards coinci-

dence, the secant ultimately becomes a tangent to the circle: we

infer therefore that a tangent cannot meet a circle otherwise than

at its point of contact. Thus Euclid's definition of a tangent may be

deduced from that given by the Method of Limits.

214

KLCLIDm t'.I.K.MKM>.

2. By this Method Proposition 32 may he derived us a special case

from rropositio)i 21.

For let A and B be two points on the O"

of the Â©ABC;

and let BCA, BPA be any two angles in

the segment BCPA :

then the z BPA^^^tho z BCA. in. 21.

Produce PA to Q.

Now let the point P continually approach

the fixed point A, and ultimately coincide

with it ;

then, however near P may approach to A,

the Z BPQ=rthe z BCA. iii. 21.

But in the limiting position when

P coincides with A,

and the secant PAQ becomes the tangent AQ',

it is clear that BP will coincide with BA,

and the Z BPQ becomes the z BAQ'.

Hence the z BAQ' = the z BCA, in the alternate segment. q. k. d.

The contact of circles may be treated in a similar manner by

adopting the following definition.

Definition. If one or other of two intersecting circles alters its

position in such a way that the two points of intersection continually

approach one another, and ultimately coincide ; in the limiting posi-

tion they are said to toucli one another, and the point in which the

two points of intersection ultimately coincide is called the point of

contact.

EXAMPLES ON LIMITS.

1. Deduce Proposition 19 from the Corollary of Proposition 1

and Proposition 3.

2. Deduce Propositions 11 and 12 from Ex. 1, page 156.

3. Deduce Proposition G from Proposition 5.

4. Deduce Proposition 13 from Proposition 10.

5. Shew that a straight line cuts a circle in two different points,

TWO coincident points, or not at all, according as its distance from the

centre is less than, equal to, or greater than a radiiis.

6. Deduce Proposition 32 from Ex. 3, page 188.

7. Deduce Proposition 36 from Ex. 7, page 209.

8. The -angle in a semi-circle is a rifjht angle.

To what Theorem is this statement reduced, when the vertex of

the right angle is brought into coincidence with an extremity of the

diameter?

9. From Ex. 1, page 190, deduce the corresponding property of a

triangle inscribed in a circle.

THEOREMS AND EXAMPLES ON BOOK III. 215

THEOREMS AND EXAMPLES ON BOOK III.

I. ON THE CENTRE AND CHORDS OF A CIRCLE.

See Propositions 1, 3, 14, 15, 25.

1. All circles tvhich j^ass through a fixed point, and have then-

centres on a (jlven straight line, pass also throxigh a second fixed point.

Let AB be the given st. line, and P the given point.

P'

From P draw PR perp. to AB ;

and produce PR to P', making RP' equal to PR.

Then all circles which pass through P, and have their centres on

AB, shall pass also through P'.

For let C be the centre of amj one of these circles.

Join CP, CP'.

Then in the a" CRP, CRP'

{CR is common,

and RPâ€” RP', Constr.

and the z CRP = the z CRP', being rt. angles ;

.-. CP = CP'; 1.4.

.'. the circle whose centre is C, and which passes through P, must

pass also through P'.

But C is the centre of any circle of the system ;

.*. all circles, which pass through P, and have their centres in AB,

pass also through P'. q. e. d.

2. Describe a circle that shall pass through three given points not

in the same straight line.

216 KUCLID'h Kr,KMKNTS.

3. Describe a circle that shall pass through two given points and

have its centre in a given straight line. When is this impossible?

4. Describe a circle of given radius to pass through two given

points. When is this impossible?

5. ABC is an isosceles triangle ; and from the vertex A, as centre,

a circle is described cutting the base, or the base produced, at X and Y.

Shew that BX = CY.

6. If two circles which intersect are cut by a straight line

parallel to the common chord, shew that the parts of it intercepted

between the circumferences are equal.

7. If two circles cut one another, any two straight lines drawn

through a point of section, making equal angles with the common

chord, and terminated by the circumferences, are equal. [Ex. 12,

p. 156.]

8. If two circles cut one another, of all straight lines drawn

through a point of section and terminated by the circumferences, the

greatest is that which is parallel to the line joining the centres.

9. Two circles, whose centres are C and D, intersect at A, B;

and through A a straight line PAQ is drawn terminated by the

circumferences: if PC, QD intersect at X, shew that the angle PXQ

is equal to the angle CAD.

10. Through a point of section of two circles which cut one

another draw a straight line terminated by the circumferences and

bisected at the point of section.

11. AB is a fixed diameter of a circle, whose centre is C; and

from P, any point on the circumference, PQ is drawn perpendicular

to AB; shew that the bisector of the angle CPQ always intersects the

circle in one or other of two fixed points.

12. Circles are described on the sides of a quadrilateral as

diameters: shew that the common chord of any two consecutive

circles is parallel to the common chord of the other two. [Ex. 9,

p. 97.]

13. Two equal circles touch one another externally, and through

the point of contact two chords are drawn, one in each circle, at

right angles to each other : shew that the straight line joining their

other extremities is equal to the diameter of either circle.

14. Straight lines are drawn from a given external point to the

circumference of a circle : find the locus of their middle points.

[Ex. 3, p. 97.]

15. Two equal segments of circles are described on opposite sides

of the same chord AB ; and through O, the middle point of AB, any

straight line POQ is drawn, intersecting the arcs of the segments at

P and Q : shew that O P - OQ.

theor?:ms and examples on book iir. 217

II. ON THE TANGENT AND THE CONTACT OP CIRCLES.

See Propositions 11, 12, 16, 17, 18, 19.

1. All equal chords placed in a given circle touch a fixed concen-

tric circle.

2. If from an external point two tangents are drawn to a circle,

the angle contained by them is double the angle contained by the

chord of contact and the diameter drawn through one of the points of

contact.

3. Two circles touch one another externally, and through the

point of contact a straight line is drawn terminated by the circun>

ferences : shew that the tangents at its extremities are parallel.

4. Two circles intersect, and through one point of section any

straight line is drawn terminated by the circumferences : shew that

the angle between the tangents at its extremities is equal to the angle

between the tangents at the point of section.

5. Shew that two parallel tangents to a circle intercept on any

third tangent a segment which subtends a right angle at the centre.

6. Two tangents are drawn to a given circle from a fixed external

point A, and any third tangent cuts them produced at P and Q: shew

that PQ subtends a constant angle at the centre of the circle.

7. In any quadrilateral circumscribed about a circle, the suvi of

one pair of opposite sides is equal to the sum of the other pair.

8. If the sum of one pair of opposite sides of a quadrilateral is

equal to the sum of the other pair, shew that a circle may be inscribed

in the figure.

[Bisect two adjacent angles of the figure, and so describe a circle to

touch three of its sides. Then prove indirectly by means of the last

exercise that this circle must also touch the fourth side.]

9. Two circles touch one another internally: shew that of all

chords of the outer circle which touch the inner, the greatest is that

which is perpendicular to the straight line joining the centres.

10. In a right-angled triangle, if a circle is described from the

middle point of the hypotenuse as centre and with a radius equal to

half the sum of the sides containing the right angle, it will touch

the circles described on these sides as diameters.

11. Through a given point, draw a straight line to cut a circle,

so that the part intercepted by the circumference may be equal to a

given straight line.

In order that the problem may 6e possible, between what limits

must the given line lie, when the given point is (i) without the circle,

(ii) within it?

218 kuclid'b klkmknts.

12. A Belies of circles touch a given straight line at a given point:

shew that the tangents to them at the points where they cut a given

parallel straight line all touch a fixed circle, whose centre is the given

point.

13. If two circles touch one another internally, and any third

circle be described touching both ; then the sum of the distances of

the centre of this third circle from the centres of the two given circles

is constant..

14. Find the locus of points such that the pairs of tangents

drawn from them to a given circle contain a constant angle.

15. Find a point such that the tangents drawn from it to two

given circles may be equal to two given straight lines. When is this

impossible?

16. If three circles touch one another two and two ; prove that

the tangents drawn to them at the three points of contact are con.

current and equal.

The Common Tangents to Two Circles.

17. 'To draw a common tangent to two circles.

First, if the given circles are external to one another, or if they

intersect.

Let A be the centre of the

greater circle, and B the centre

of the less.

From A, with radius equal

to the diff^** of the radii of the

given circles, describe a circle:

and from B draw BC to touch

the last di'awn circle. Join AC,

and produce it to meet the

greater of the given circles at D.

Through B draw the radius BE pai^ to AD, and in the same

direction.

Join DE:

then DE shall be a common tangent to the two given circles.

For since AC = the diff'Â« between AD and BE, Comtr.

.'. CD=BE:

and CD is par^ to BE; Constr.

.-. DE is equal and par' to CB. i. 33.

But since BC is a tangent to the circle at C,

.-. the z ACB is a rt. angle; in. 18,

hence each of the angles at D and E is a rt. angle: i. 29.

.â€¢. DE is a tangent to both circles. q.e.f.

THEOREMS AND EXAMPLES ON BOOK III. 219

It follows from hypothesis that the point B is outside the circle

used in the construction :

.-. two tangents such as BC may always be drawn to it from B ;

hence two common tangents may always be drawn to the given

circles by the above method. These are called the direct common

tangents.

When the given circles are external to one another and do not

intersect, two more common tangents may be drawn.

For, from centre A, with a radius equal to the siuii of the radii of

the given circles, describe a circle.

From B draw a tangent to this circle ;

and proceed as before, but draw BE in the direction opposite to AD.

It follows from hypothesis that B is external to the circle used in

the construction ;

.-. two tangents may be drawn to it from B.

Hence tioo more conmion tangents may be drawn to the given

circles : these will be found to pass between the given circles, and are

called the transverse common tangents.

Thus, in general, four common tangents may be drawn to two

given circles.

The student should investigate for himself the number of common

tangents which may be drawn in the following special cases, noting

in each case where the general construction fails, or is modified : â€”

(i) When the given circles intersect :

(ii) When the given circles have external contact :

(iii) When the given circles have internal contact :

(iv) When one of the given circles is wholly within the other.

18. Drmo the direct common tangents to two equal circles.

19. If the two direct, or the two transverse, common tangents

are drawn to two circles, the parts of the tangents intercepted be-

tween the points of contact are equal.

20. If four common tangents are drawn to two circles external to

one another; shew that the two direct, and also the two transverse,

tangents intersect on the straight line which joins the centres of the

circles.

21. Two given circles have external contact at A, and a direct

common tangent is drawn to touch them at P and Q. : shew that PQ

subtends a right angle at the point A.

22. Two circles have external contact at A, and a direct common

tangent is drawn to touch them at P and Q: shew that a circle

described on PQ as diameter is touched at A by the straight line

which joins the centres of the circles.

220 EDCLTD's KliEMENTS.

23. Two circles whose centres are C and C have external contact

at A, and a direct common tangent is drawn to touch them at P

and Q : shew that the bisectors of the angles PCA, QC'A meet at

right angles in PQ. And if R is the point of intersection of the

bisectors, shew that RA is also a common tangent to tlie circles.

24. Two circles have external contact at A, and a direct common

tangent is drawn to touch them at P and Q : shew that the square

on PQ is equal to the rectangle contained by the diameters of the

circles.

25. Draw a tangent to a given circle, so that the part of it

intercepted by another given circle may be equal to a given straight

line. When is this impossible?

26. Draw a secant to two given circles, so that the parts of it

intercepted by the circumferences may be equal to two given straight

lines.

Problems on Tangency.

The following exercises are solved by the Method of Inter-

section of Loci, explained on page 117.

The student should begin by making himself familiar with

the following loci.

(i) The loais of the centres of circles which pass through two given

points.

(ii) The locus of the centres of circles xohich touch a given straight

line at a given point.

(iii) The locus of the centres of circles tchich touch a given circle at

a given point.

(iv) The locus of the centres of circles ivhich touch a given straight

line, and have a given radius.

(v) 'The locus of the centres of circles tchich touch a given circle,

and have a given radius.

(vi) 'The locus of the centres of circles which touch two given

straight lines.

In each exercise the student should investigate the limits

and relations among the data, in order that the problem may be

possible.

27. Describe a circle to touch three given straight lines.

28. Describe a circle to pass through a given point and touch a

given straight line at a given point.

29. Describe a circle to pass through a given point, and touch a

given circle at a given point.

THEOREMS AND EXAMPLES ON BOOK 111. 22 i

30. Describe a circle of given radius to pass through a given

point, and touch a given straight line.

31. Describe a circle of given radius to touch two given circles.

32. Describe a circle of given radius to touch two given straight

lines.

33. Describe a circle of given radius to touch a given circle and a

given straight line.

34. Describe two circles of given radii to touch one another and

a given straight line, on the same side of it.

35. If a circle touches a given circle and a given straight line,

shew that the points of contact and an extremity of the diameter of

the given circle at right angles to the given line are collinear.

36. To describe a circle to touch a given circle, and aho to touch a

given straight line at a given point.

Let DEB be the given circle, PQ.

the given st. line, and A the given

point in it :

it is required to describe a circle to

touch the Â© DEB, and also to touch

PQ at A.

At A draw AF perp. to PQ : 1. 11.

then the centre of the required circle

must lie in AF. iii. 19.

Find C, the centre of the Â© DEB,

I. 1.

and draw a diameter BD perp. to

PQ:

join A to one extremity D, cutting

the o"'' at E.

Join CE, and produce it to cut AF at F.

Then F is the centre, and FA the radius of the required circle.

[Supply the proof : and shew that a second solution is obtained by

joining AB, and producing it to meet the O"" :

also distinguish between the nature of the contact of the circles, when

PQ cuts, touches, or is without the given circle.]

37. Describe a circle to touch a given straight line, and to touch

a given circle at a given point.

38. Describe a circle to touch a given circle, have its centre in a

given straight line, and pass through a given point in that straight

line.

[For other problems of the same class see page 235.]

222 EUCLID'S KLEMENTS.

Oktuogonal Chicles.

Definition. Circles which intersect .'it a- point, so that the

two tangents at that point are at right angles to one another,

are said to be orthogonal, or to cut one another ortho-

gonally.

39. In two intersectiug circles the angle between the ta*)gents

at one point of intersection is equal to the angle between the tangents

at the other.

40. 7/ two circles cut one another orlJiononully, the tangent io

each circle at a point of intersection will p(/,sÂ« throwjh the centre of

the other circle.

41. If tico circles cut one another orthogonally, the nquare on the

distance between their centres is equal to the sum of the squares on

their radii.

42. Find the locus of the centres of all circles which cut a given

circle orthogonally at a given point.

43. Describe a circle to pass through a given point and cut a

given circle orthogonally at a given point.

III. ON ANGLES IN SEGMENTS, AND ANGLES AT THE

CENTRES AND CIRCUMFERENCES OF CIRCLES.

See Propositions 20, 21, 22; 26, 27, 28, 29; 31, 32, 33, 34.

1. If two chords intersect to i thin a circle, they form an angle equal

to that at the centre, subtended by half the sum of the arcs they cut off.

Let AB and CD be two chords, intersecting

at E within the given Â©ADBC:

then shall the l A EC be equal to the angle at

the centre, subtended by half the sum of the

arcs AC, BD.

Join AD.

Then the ext. z A EC = the sum of the int.

opp. /"EDA, EAD;

that is, the sum of the z ' CDA, BAD.

But the z-'CDA, BAD are the angles at

the O'^'' subtended by the arcs AC, BD ;

.-. their sum = half the sum of the angles at the centre subtended by

the same arcs;

or, the z A EC = the angle at the centre subtended by half the sum of

the arcs AC, BD. q. e. d.

THEOREMS AND EXAMPLES ON BOOK III. 223

2. If tivo chords lohen produced intersect outside a circle, they form

an angle equal to that at the centre subtended by half the difference of

the arcs they cut of.

B. The sum of the arcs cut off by two chords of a circle at right

angles to one another is equal to the semi-circumference.

4. AB, AC are any two chords of a circle; and P, Q are the

middle points of the minor arcs cut off by them: if PGl is joined,

cutting AB and AC at X, Y, shew that AX = AY.

5. If one side of a quadrilateral inscribed in a circle is inoduced,

the exterior angle is equal to the opposite interior angle.

6. If two circles intersect, and any straight lines are drawn, one

through each point of section, terminated by the circumferences;

shew that the chords which join their extremities towards the same

parts are parallel.

7. ABCD is a quadrilateral inscribed in a circle ; and the opposite

sides AB, DC are produced to meet at P, and CB, DA to meet at Q:

if the circles circumscribed about the triangles PBC, QAB intersect

at R, shew that the points P, R, Q are collinear.

8. If a circle is described on one of the sides of a right-angled

triangle, then the tangent drawn to it at the point where it cuts the

hypotenuse bisects the other side.

9. Given three points not in the same straight line : shew how

to find any uumber of points on the circle which passes through them,

without finding the centre.

10. Through any one of three given points not in the same

straight line, draw a tangent to the circle which passes through them,

without finding the centre.

11. Of two circles which intersect at A and B, the circumference

of one passes through the centre of the other : from A any straight

line ACD is drawn to cut them both; shew that CB = CD.

12. Two tangents AP, AQ are drawn to a circle, and B is the

middle point of the arc PQ, convex to A. Shew that PB bisects the

angle APQ.

13. Two circles intersect at A and B; and at A tangents are

drawn, one to each circle, to meet the circumferences at C and D : if

CB, BD are joined, shew that the triangles ABC, DBA are equi-

angular to one another.

14. Two segments of circles are described on the same chord and

on the same side of it ; the extremities of the common chord are joined

to any point on the arc of the exterior segment : shew that the arc

intercepted on the interior segment is constant.

224 Euclid's elements.

15. If a series of triangles are drawn standing on u lixed basi',

and having a given vertical angle, shew that the bisectors of the verti-

cal angles all pass through a fixed point.

16. ABC is a triangle inscribed in a circle, and E the middle

point of the arc subtended by BC on the side remote from A: ii"

through E a diameter ED is drawn, shew that the angle DEA is half

the difference of the angles at B and C. [See Ex. 7, p. 101.]

17. If two circles touch each other internally at a point A, any

chord of the exterior circle which touches the interior is divided at its

point of contact into segments which subtend equal angles at A.

18. If two circles touch one another internally, and a straight

line is drawn to cut them, the segments of it intercepted between the

circumferences subtend equal angles at the point of contact.

The Orthocen'tre of a Triangle.

I'J. The perpendiculars draini from the vertices of a Iriatiyle to

the opposite sides are concurrent.

In the A ABC, let AD, BE be the

perp' drawn from A and B to the oppo-

site sides ; and let them intersect at O,

Join CO; and produce it to meet AB

at F.

It is required to shew that CF is perp.

to AB.

Join DE.

Then, because the z ' OEC, GDC are

rt. angles, Hijp.

:. the points O, E, C, D are concyclic :

.-. the z DEC = the Z DOC, in the same segment;

= the vert. opp. Z FOA.

Again, because the z " AEB, ADB are rt. angles, Hyp.

:. the points A, E, D, B are concyclic:

.-. the Z DEB = the Z DAB, in the same segment.

.-. the sum of the Z " FOA, FAO = the sum of the z ' DEC, DEB

= a rt. angle : ^VP-

:. the remaining Z AFO = art. angle: i. 32,

that is, CF is perp. to AB.

Hence the three perp' AD, BE, CF meet at the point O. q. e. d.

[For an Alternative Proof see page lOG.]

THEOKKMS AND EXAMPLES ON BOOK III. 225

Definitions.

(i) The intersection of the perpendiculars drawn from the

vortices of a triangle to the opposite sides is called its ortho-

centre.

(ii) The triangle formed by joining the feet of the perpendi-

culars is called the pedal or orthocentric triangle.

20. In an acute-angled triangle the perpendiculars draivn from

the vertices to the opposite sides bisect the angles of the pedal triangle

through ivhich they pass.

In the acute-angled a ABC, let AD,

BE, CF be the perj)^ drawn from the

vertices to the opposite sides, meeting

at the orthocentre O; and let DEF be

the pedal triangle :

then shall AD, BE, CF bisect respect-

ively the z Â« FDE, DEF, EFD.

For, as in the last theorem, it may

be shewn that the points O, D, C, E are

concyclic ;

.â€¢. the Z ODE = the Z OCE, in the same segment.

Similarly the points O, D, B, F are concyclic;

,â€¢. the z ODF â€” the z OBF, in the same segment.

But the z OCE = the z OBF, each being the comp' of the z BAC.

.-. the Z ODE = the z ODF.

Similarly it may be shewn that the z " DEF, EFD are bisected by

BE and CF. q. e.d.

Corollary, (i) Every two sides of the pedal triangle are equally

inclined to that side of the original triangle in icliich they meet.

For the z EDC = the comp* of the Z ODE

= the comp' of the Z OCE

= the Z BAC.

Similarly it may be shewn that the z FDB = the Z BAC,

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