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Euclid.

Euclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools online

. (page 17 of 38)
Online LibraryEuclidEuclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools → online text (page 17 of 38)
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angle,

and therefore Ci^is perpendicular to AD.
Synthesis. From the center C, draw CF perpendicular to AD
meeting the circumference of the circle in F:

join DF cutting the circumference in E,
join also CE, and at E draw EG perpendicular to CE and inter-
secting BD in G.

Then G will be the point required.
For in the triangle CFD, since FCD is a right angle, the angles
CFD, CDF are together equal to a right angle ;

also since CEG is a right angle,
therefore the angles CEF, GED are together equal to a right

angle ;
therefore the angles CEF, GED are equal to the angles CFD,
CDF',

but because CE is equal to CF,
the angle CEF is equal to the angle CFD,
wherefore the remaining angle GED is equal to the remaining

angle CDF,
and the side GD is equal to the side GE of the triangle EGD,
therefore the point G is determined according to the required
conditions.

PROPOSITION III. THEOREM.

If a chord of a circle be produced till the part prodttced be equal to the
radius, and if from its extremity a lijie be drawn through the center and
meeting the convex and concave circumferences, the convex is one-third of the
concave circumference.

Let AB any chord be produced to C, so that BC \& equal to the
radius of the circle :




and let CE be drawn from C through the center D, and meeting
the convex circumference in F, and the concave in E.
Then the arc BF is one-third of the arc AE.



162 GEOMETRICAL EXERCISES

Draw EG parallel to AB, and join DB, DG.

Since the angle DEG is equal to the angle DGE-, (i. 5.)

and the angle GDFk equal to the angles BEG, DGE-, (l. 32.)

therefore the angle GECis double of the angle DEG.

But the angle BECis equal to the angle BCD, (i. 5.)

and the angle CEG is equal to the alternate angle A CE ; (l. 29.)

therefore the angle GDCis double of the angle CDB,

add to these equals the angle CDB,

therefore the whole angle GDB is treble of the angle CDB,

but the angles GDB, CDB at the center D, are subtended by the

arcs BE, BG, of which BG is equal to ^^.
Wherefore the circumference AE is treble of the circumference
BE, and BE is one-third of ^^.

Hence may be solved the following problem :
AE, BE are two arcs of a circle intercepted between a chord and
a given diameter. Determine the position of the chord, so that one
arc shall be triple of the other.

PROPOSITION IV. THEOREM.

AB, AC and ED are tangents to the circle CFB; at whatever point
between C and B the tangent EFD is drawn, the three sides of the triangle
AED are equal to twice AB or twice AC: also the angle subtended by the
tangent EFD at the center of the circle^ is a constant quantity.

Take G the center of the circle, and join GB, GE, GF, GD, GC
Then EB is equal to EF, and DC to DF; (iii. 37.)




therefore ED is equal to EB and DC;

to each of these add AE, AD,

wherefore AD, AE, ED are equal to AB, AC;

and AB is' equal to AC,

therefore AD, AE, ED are equal to twice AB, or twice A C\

or the perimeter of the triangle AED is a constant quantity.

Again, the angle EGF is half of the angle BGF,

and the angle DGE is half of the angle CGF,

therefore the angle DGE is half of the angle CGB,

or the angle subtended by the tangent ED2X G,is half of the angle

contained between the two radii which meet the circle at the points

where the two tangents AB, ^Cmeet the circle.

PROPOSITION V. PROBLEM.

Given the base, the vertical angle, and the perpendicular in a plane triangle,
to construct it.



r

■r Udoq the )



ON BOOK III.



163



Upon the given base AJB describe a segment of a circle containing
an angle equal to the given angle, (in. 33.)



I




At the point B draw ^C perpendicular to AJB, and equal to the
altitude of the triangle. (l. 11, 3.)

Through C, draw CDJE parallel to AB, and meeting the circum-
ference in D and JE. (i. 31.)

Join JDA, DB ; also BA, EB ;
then EAB or DAB is the triangle required.
It is also manifest, that if CDE touch the circle, there will be only
one triangle which can be constructed on the base AB with the given
altitude.

PROPOSITION YI. THEOREM.

If two chords of a circle intersect each other at right angles either within or
without the circle, the sum of the squares described upon the four segments, is
equal to the square described upon the diameter.



Let the chords AB, CD intersect at right angles in E,

A




leone



I

"|^» Draw the diameter AF, and join A C, AD, CF, DB.

Then the angle ACF in a semicircle is a right angle, (in. 31.)

and equal to the angle AED :

also the angle ADC is equal to the angle AFC (in. 21.)

Hence in the triangles ADE, AFC, there are two angles in the

respectively equal to two angles in the other,

consequently, the third angle CAF is equal to the third angle
DAB',

therefore the arc DB is equal to the arc CF, (in. 26.)

and therefore also the chord DB is equal to the chord CF. (in. 29.)

Because AECis a right-angled triangle,

the squares on AE, EC are equal to the square on A C; (l. 47.)

similarly, the squares on DE, EB are equal to the square on DB ;

therefore the squares on AE, EC, DE, EB, are equal to the squares

on AC, DBi

I but DB was proved equal to FC,



164 GEOMETRICAL EXERCISES

wherefore the squares on AE, EC, DE, EB, are equal to the square

on AF, the diameter of the circle.
When the chords meet without the cii'cle, the property is proved

in a similar manner.



I.

7. Theougii a given point within a circle, to draw a chord which
shall be bisected in that point, and prove it to be the least.

8. To draw that diameter of a given circle which shall pass at a
given distance from a given point.

9. Find the locus of the middle points of any system of parallel
chords in a circle.

10. The two straight lines which join the opposite extremities of
two parallel chords, intersect in a point in that diameter which is
perpendicular to the chords.

11. The straight lines joining towards the same parts, the extre-
mities of any two lines in a circle equally distant from the center, are
parallel to each other.

12. A, B, C, A', B', C are points on the circumference of a circle ;
if the lines AB, AC he respectively parallel to A'B ^ A'C'^ shew that
jBC" is parallel to ^' a

13. Two chords of a circle being given in position and magnitude,
describe the circle.

14. Two circles are drawn, one lying within the other ; prove that
no chord to the outer circle can be bisected in the point in which it
touches the inner, unless the circles are concentric, or the chord be
perpendicular to the common diameter. If the circles have the same
center, shew that every chord which touches the inner circle is bisected
in the point of contact.

15. Draw a chord in a circle, so that it may be double of its per-
pendicular distance from the center.

16. The arcs intercepted between any two parallel chords in a circle
are equal.

17. If any pt)int P be taken in the plane of a circle, and PAj
PB, PC,.. he di-awn to any number of points A, B, C, .. situated
sj-Tnmetrically in the circumference, the sum of PA^ Pi?,.. is least
when P is at the center of the circle.

II.

18. The sum of the arcs subtending the vertical angles made by
any two chords that intersect, is the same, as long as the angle of inter-
section is the same.

19. From a point without a circle two straight lines are drawn
cutting the convex and concave circumferences, and also respectively
parallel to two radii of the circle. Prove that the d.ifFerence of the
concave and convex arcs intercepted by the cutting lines, is equal to
twice the arc intercepted by the radii.

20. In a circle with center O, any two chords, AB, CD are drawn



ON BOOK III. 165-

cutting In E, and OA, OB, OC, OD are joined ; pro-ve that the angles
AOC+ JB0D = 2.AI:Q and A0I) + JB0C^2.A£:i).

21. If from any point without a circle, lines be di'awn cutting the
circle and making equal angles with the longest line, they will cut off
equal segments.

22. If the corresponding extremities of two intersecting chords of
a circle be joined, the triangles thus formed will be equiangular.

23. Through a given point within or without a circle, it is required
to draw a sti-aight line cutting off a segment containing a given angle.

24. If on two lines containing an angle, segments of circles be
described containing angles equal to it, the lines produced will touch
the segments.

25. Any segment of a circle being described on the base of a tri-
angle ; to describe on the other sides segments similar to that on the
base.

26. If an arc of a circle be divided into three equal parts by three
straight lines drawn from one extremity of the arc, the angle con-
tained by two of the straight lines is bisected by the third.

27. If the chord of a given circular segment be produced to a
fixed point, describe upon it when so produced a segment of a circle
which shall be similar to the given segment, and shew that the two
segments have a common tangent.

28. If ^Z>, CU be drawn perpendicular to the sides BC, AB of
the triangle ABC, and DU be joined, prove that the angles ADE,
and A CE are equal to each other.

29. If from any point in a circular arc, perpendiculars be let fall
on its bounding radii, the distance of their feet is invariable.

III.

30. If both tangents be drawn, (fig. Euc. Til. 17.) and the points
of contact joined by a straight line which cuts EA in JI, and on HA
as diameter a circle be described, the lines drawn through E to touch
this circle will meet it on the circumference of the given circle.

31. Draw, (1) perpendicular, (2) parallel to a given line, a line
touching a given circle.

32. If two straight lines intersect, the centers of all circles that
can be inscribed between them, lie in two lines at right angles to each
other.

33. Draw two tangents to a given circle, which shall contain an
angle equal to a given rectilineal angle.

34. Describe a circle with a given radius touching a given line, and
so that the tangents drawn to it from two given points in this line
may be parallel, and shew that if the radius vary, the locus of the
centers of the circles so described is a circle.

35. Determine the distance of a point from the center of a given
circle, so that if tangents be drawn from it to the circle, the concave
part of the circumference may be double of the convex.

36. In a chord of a circle produced, it is required to find a point,
from which if a straight line be drawn touching the circle, the line so
drawn shall be equal to a given straight line.



166 GEOMETRICAL EXERCISES

37. Find a point without a given circle, such that the sum of the
two lines drawn from it touching the circle, shall be equal to the line
drawn from it through the center to meet the circle.

38. If from a point without a circle two tangents be drawn ; the
straight line which joins the points of contact will be bisected at right
angles by a line drawn from the center to the point without the circle.

39. If tangents be drawn at the extremities of any two diameters
of a circle, and produced to intersect one another ; the straight lines
joining the opposite points of intersection will both pass through
the center.

40. If from any point without a circle two lines be drawn touching
the circle, and from the extremities of any diameter, lines be drawn to
the point of contact cutting each other within the circle, the line di'awn
from the points without the cii'cle to the point of intersection, shall be
perpendicular to the diameter.

41. If any chord of a circle be produced equally both ways, and
tangents to the circle be drawn on opposite sides of it from its extre-
mities, the line joining the points of contact bisects the given chord.

42. AB is a chord, and AD is a tangent to a circle at A. DFQ
any secant parallel to AJB meeting the circle in P and Q. Shew that
the triangle FAD is equiangular with the triangle QAB.

43. If from any point in the circumference of a circle a chord and
tangent be drawn, the perpendiculars dropped upon them from the
middle poin^of the subtended arc, are equal to one another.

IV.

44. In a given straight line to find a point at which two other
straight lines being drawn to two given points, shall contain a right
angle. Shew that if the distance between the tAvo given points be
greater than the sum of their distances from the given line, there will
be two such points; if equal, there may be only one; if less, the
problem may be impossible.

45. Find the point in a given straight line at which the tangents
to a given circle will contain the greatest angle.

46. Of all straight lines which can be di'awn from two given points
to meet in the convex circumference of a given circle, the sum of those
two will be the least, which make equal angles with the tangent at the
point of concourse.

47. DF is a straight line touching a circle, and terminated by
AD, BF, the tangents at the extremities of the diameter AB, shew
that the angle which i)i^ subtends at the center is a right angle.

48. If tangents Am, Bn be drawn at the extremities of the dia-
meter of a semicircle, and any line in mFn crossing them and touching
the circle in P, and if AN, B3I be joined intersecting in O and cutting
the semicircle in F and F; shew that O, P, and the point of intersec-
tion of the tangents at F and F, are in the same straight line.

49. If from a point P without a circle, any straight line be drawn
cutting the circumference in A and B, shew that the straight lines
joining the points A and B with the bisection of the chord of contact
of the tangents from P, make equal angles with that chord.




ON BOOK III. 167

_ V.

50. Describe a circle which shall pass through a given point and
M'hich shall touch a given straight line in a given point.

51. Draw a straight line which shall touch a given circle, and
make a given angle with a given straight line.

52. Describe a circle the circumference of which shall pass through
a given point and touch a given circle in a given point.

53. Describe a circle with a given center, such that the circle so
described and a given circle may touch one another internally.

54. Describe the circles which shall pass through a given point
and touch two given straight lines.

55. Describe a circle with a given center, cutting a given circle in
the extremities of a diameter.

56. Describe a circle which shall have its center in a given straight
line, touch another given line, and pass through a fixed point in the
first given line.

57. The center of a given circle is equidistant from two given
straight lines ; to describe another circle which shall touch the two
straight lines and shall cut off from the given circle a segment con-
taining an angle equal to a given rectilineal angle.

VI.

58. If any two circles, the centers of which are given, intersect
each other, the greatest line which can be drawn through either point
of intersection and terminated by the circles, is independent of the
diameters of the circles.

59. Two equal circles intersect, the lines joining the points in
which any straight line through one of the points of section, which
meets the circles with the other point of section, are equal.

60. Draw through one of the points in which any two circles cut
one another, a straight line which shall be terminated by their circum-
ferences and bisected in their point of section.

61. Describe two circles with given radii which shall cut each
other, and have the line between the points of section equal to a given
line.

62. Two circles cut each other, and from the points of intersection
straight lines are drawn parallel to one another, the portions inter-
cepted by the circumferences are equal.

63. ACB, ADB are two segments of circles on the same base
AB, take any point Cin the segment ACB; join AC, BC, and pro-
duce them to meet the segment ADB in D and JE respectively : shew
that the arc DE is constant.

64. ADB, ACB, are the arcs of two equal circles cutting one
another in the straight line AB, draw the chord ^ CD cutting the
inner circumference in (7 and the outer in D, such that AD and DB
together may be double of ^C and CB together,

65. If from two fixed points in the circumference of a circle,
straight lines be drawn intercepting a given arc and meeting without
the circle, the locus of their intersections is a circle.



168 GEOMETRICAL EXERCISES

66. If two circles intersect, the common chord produced bisects
the common tangent.

67. Shew that, if two circles cut each other, and from any point
in the straight line produced, which joins their intersections, two tan-
gents be drawn, one to each circle, they shall be equal to one another.

68. Two circles intersect in the points A and B : through A and
J5 any two straight lines CAB, EBF, are drawn cutting the cii-cles in
the points C, D, E, F-, prove that CE is parallel to DF.

69. Two equal circles are drawn intersecting in the points A and
B, a third circle is drawn with center A and any radius not greater
than AB intersecting the former circles in D and C. Shew that the
three points, B, C, I) lie in one and the same straight line.

70. If two circles cut each other, the straight line joining their
centers will bisect their common chord at right angles.

71.^ Two circles cut one another ; if through a point of intersection
a straight line is drawn bisecting the angle betM-een the diameters at
that point, this line cuts off similar segments in the two circles.

72. ACB, APB are two equal circles, the center of APB being
on the circumference of ACB, AB being the common chord, if any
chord ^ C of ^ C^ be produced to cut ABB in P, the triangle PBC
is equilateral.

VII.

73. If two circles touch each other externally, and two parallel
lines be drawn, so touching the circles in points A and B respectively
that neither circle is cut, then a straight line AB will pass through
the point of contact of the circles.

74. A common tangent is di-awn to two circles w^hich touch each
other externally ; if a circle be described on that part of it which lies
between the points of contact, as diameter, this circle will pass through
the point of contact of the two circles, and will touch the line which
joins their centers.

75. If two cu'cles touch each other externally or internally, and
parallel diameters be drawn, the straight line joining the extremities
of these diameters will pass through the point of contact.

76. If two circles touch each other internally, and any circle be
described touching both, prove that the sum of the distances of its
center from the centers of the two given circles will be invariable.

77. If two circles touch each other, any straight line passing
through the point of contact, cuts off similar parts of their circumfe-
rences.

78. Two circles touch each other externally, the diameter of one
being double of the diameter of the other ; through the point of con-
tact any line is drawn to meet the circumferences of both ; shew that
the part of the line w^hich lies in the larger circle is double of that in
the smaller.

79. If a circle roll within another of twice its size, any point in
its circumference will trace out a diameter of the first.

80. With a given radius, to describe a circle touching two given
circles.



I



ON BOOK III. 169



81. Two equal circles touch one another externally, and through
the point of contact chords are drawn, one to each circle, at right
angles to each ; prove that the straight line joining the other extre-
mities of these chords is equal and parallel to the straight Hue joining
the centres of the circles.

82. Two circles can be described, each of which shull touch a
given circle, and pass through two given points outside the circle ;
shew that the angles which the two given points subtend at the two
points of contact, are one greater and the other less than that which
they subtend at any other point in the given circle.



vm.

83. Draw a straight line which shall touch two given circles;
(1) on the same side ; (2) on the alternate sides.

84. If two circles do not touch each other, and a segment of the
line joining their centers be intercepted between the convex circum-
ferences, any circle whose diameter is not less than that segment may
be so placed as to touch both the circles.

85. Given two circles : it is required to find a point from which
tangents may be drawn to each, equal to two given straight lines.

86. Two circles are traced on a plane ; draw a straight line
cutting them in such a manner that the chords intercepted within the
circles shall have given lengths.

87. Draw a straight line which shall touch one of two given circles
and cut off a given segment from the other. Of how many solutions
does this problem admit ?

88. If from the point where a common tangent to two circles
meets the line joining their centers, any line be drawn cutting the
circles, it will cut off similar segments.

89. To find a point P, so that tangents drawn from it to the out-
sides of two equal circles which touch each other, may contain an angle
equal to a given angle.

90. Describe a circle which shall touch a given straight line at a
given point, and bisect the circumference of a given circle.

91. A circle is described to pass through a given point and cut a
given circle orthogonally, shew that the locus of the center is a certain
straight line.

92. Through two given points to describe a circle bisecting the
circumference of a given circle.

93. Describe a circle through a given point, and touching a given
straight line, so that the chord joining the given point and point of
contact, may cut off a segment containing a given angle.

94. To describe a circle through two given points to cut a straight
line given in position, so that a diameter of the circle di'awn through
the point of intersection, shall make a given angle with the line.

95. Describe a circle which shall pass through two given points
and cut a given circle, so that the chord of intersection may be of a
given length.



170 GEOMETRICAL EXERCISES



IX.



96. The circumference of one circle is wholly within that of an-
other. Find the greatest and the least straight Unes that can be drawn
touching the former and terminated by the latter.

97. Draw a straight line through two concentric circles, so that the
chord terminated by the exterior circumference may be double that
terminated by the interior. What is the least value of the radius of
the interior circle for which the problem is possible ?

98. If a straight line be drawn cutting any number of concentric
circles, shew that the segments so cut off are not similar.

99. If from any point in the circumference of the exterior of two
concentric circles, two straight lines be drawn touching the interior
and meeting the exterior ; the distance bet>veen the points of contact
will be half that between the points of intersection.

100. Shew that all equal straight lines in a circle will be touched
by another circle.

101. Through a given point draw a straight line so that the part
intercepted by the circumference of a circle, shall be equal to a given
straight line not greater than the diameter.

102. Two circles are described about the same center, draw a chord
to the outer circle, which shall be divided into three equal parts by the
inner one. How is the possibility of the problem limited ?

103. Find a point without a given circle from which if two tan-
gents be drawn to it, they shall contain an angle equal to a given
angle, and shew that the locus of this point is a ckcle concentric with
the given circle.

104. Draw two concentric circles such that those chords of the
outer circle which touch the inner, may be equal to its diameter.

105. Find a point in a given straight line from which the tangent
drawn to a given circle, is of given length.

106. If any number of chords be drawn in the inner of two con-
centric circles, from the same point A in its cu'cumference, and each
of the chords be then produced beyond A to the circumference of the
outer circle, the rectangle contained by the whole line so produced
and the part of it produced, shall be constant for all the cases.

X.

107. The circles described on the sides of any triangle as diameters
will intersect in the sides, or sides produced, of the triangle.

108. The circles which are described upon the sides of a right-
angled triangle as diameters, meet the hypotenuse in the same point ;
and the line drawn from the point of intersection to the center of either
of the circles will be a tangent to the other circle.

109. If on the sides of a triangle circular arcs be described contain-
ing angles whose sum is equal to two right angles, the triangle formed



Online LibraryEuclidEuclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools → online text (page 17 of 38)