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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 ©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 RP— 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'« 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.

.•. 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 : —

(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 © 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 © 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(/,s« 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 ©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 « FDE, DEF, EFD.

For, as in the last theorem, it may
be shewn that the points O, D, C, E are
concyclic ;

.•. the Z ODE = the Z OCE, in the same segment.

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

,•. the z ODF — 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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