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

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Font size three straight lines be di-awn, making equal angles with the base, viz.
one from its extremity, the other two from any other point in it, these
two shall be together equal to the first.

28. A straight line is drawn, terminated by one of the sides of an
isosceles triangle, and by the other side produced, and bisected by
the base ; prove that the sti-aight lines, thus intercepted between the

e2

76 GEOMETRICAL EXERCISES

vertex of the isosceles triangle, and this straight line, are together
equal to the two equal sides of the triangle.

29. In a triangle, if the lines bisecting the angles at the base be
equal, the triangle is isosceles, and the angle contained by the bisect-
ing lines is equal to an exterior angle at the base of the triangle.

30. In a triangle, if lines be equal when drawn from the extremi-
ties of the base, (1) perpendicular to the sides, (2) bisecting the sides,
(3) making equal angles with the sides: the triangle is isosceles:
and then these lines which respectively join the intersections of the
sides, are parallel to the base. J

n. 5

31. AJBCis a triangle right-angled at B, and having the angle A
double the angle C; shew that the side ^C is less than double the
side AB.

32. If one angle of a triangle be equal to the sum of the other
two, the greatest side is double of the distance of its middle point from
the opposite angle.

33. If from the right angle of a right-angled triangle, two straight
lines be drawn, one perpendicular to the base, and the other bisecting
it, they will contain an angle equal to the difference of the two acute
angles of the triangle.

34. If the vertical angle CAB of a triangle ABC be bisected by
AD, to which the perpendiculars CJE, BF are drawn from the remain-
ing angles: bisect the base BCin G, join GE, GF, and prove these
lines equal to each other.

35. The difference of the angles at the base of any triangle, is
double the angle contained by a line drawn from the vertex perpen-
dicular to the base, and another bisecting the angle at the vertex.

36. If one angle at the base of a triangle be double of the other,
the less side is equal to the sum or difference of the segments of the
base made by the perpendicular from the vertex, according as the
angle is greater or less than a right angle.

37. If two exterior angles of a triangle be bisected, and from the
point of intersection of the bisecting lines, a line be drawn to the op-
posite angle of the triangle, it will bisect that angle.

38. From the vertex of a scalene triangle draw a right line to
the base, which shall exceed the less side as much as it is exceeded
by the greater.

39. Divide a right angle into three equal angles.

40. One of the acute angles of a right-angled triangle is three
times as great as the other ; trisect the smaller of these.

41. Prove that the sum of the distances of any point within
a triangle from the three angles is greater than half the perimeter
of the triangle.

42. The perimeter of an isosceles triangle is less than that of any
other equal triangle upon the same base.

43. If from the angles of a triangle ABC, straight lines AD'E,
BDF, CDG be di-awn through a point D to the opposite sides,
prove that the sides of the triangle are together greater than the three

ON BOOK I. 7*7

lines drawn to the point Z), and less than twice the same, but greater
than two-thirds of the lines drawn through the point to the opposite
sides.

44. In a plane triangle an angle is right, acute or obtuse, ac-
cording as the line joining the vertex of the angle with the middle
point of the oj^posite side is equal to, greater or less than half of
that side.

45. If the straight line AD bisect the angle A of the triangle
ABC, and BDE be drawn perpendicular to AD and meeting AC or
A C produced in E, shew that BD = DE.

46. The side BC oi a triangle ABC is produced to a point D.
The angle ACB is bisected by a line CE which meets AB in E.
A line is di-awn through E parallel to BC and meeting ^C in i^,
and the line bisecting the exterior angle A CD, in G. Shew that
ÂŁJ^is equal to i^G^.

47. The sides AB, AC, of a triangle are bisected in D and E
respectively, and BE, CD, are produced until EF= EB, and GD = DC;
shew that the line 6^i^ passes through A.

48. In a triangle ABC, AD being drawn perpendicular to the
straight line BD which bisects the angle B, shew that a line drawn
from D parallel to ^Cwill bisect AC.

49. If the sides of a triangle be trisected and lines be drawn
through the points of section adjacent to each angle so as to form
another triangle, this shall be in all respects equal to the first
ti'iangle.

50. Between two given straight lines it is required to draw a
sti-aight line which shall be equal to one given straight line, and â–
parallel to another.

51. If from the vertical angle of a triangle three straight lines be
drawn, one bisecting the angle, another bisecting the base, and the
third perpendicular to the base, the first is always intermediate in
magnitude and position to the other two.

52. In the base of a triangle, find the point from which, lines
drawn parallel to the sides of the triangle and limited by them, are equal.

53. In the base of a triangle, to find a point from which if two
lines be drawn, (1) perpendicular, (2) parallel, to the two sides of the
triangle, their sum shall be equal to a given line.

III.

54. In the figure of Euc. I. 1, the given line is produced to meet
either of the circles in P ; shew that P and the points of intersection
of the circles, are the angular points of an equilateral triangle.

55. If each of the equal angles of an isosceles triangle be one-
fourth of the third angle, and from one of them a line be drawn
at right angles to the base meeting the opposite side produced ; then
will the part produced, the perpendicular, and the remaining side,
form an equilateral triangle.

56. In the figure Euc. I. 1, if the sides CA, CB of the equilateral
triangle ABC he produced to meet the circles in F, G, respectively,
and if C be the point in which the circles cut one another on the

78 GEOMETRICAL EXERCISES

Other side of AB : prove the points F, C, G to be in the same straight
line ; and the figure CFG to be an equilateral triangle.

57. ABC is a triangle and the exterior angles at B and C
are bisected by lines BI), CD respectively, meeting in Z): shew
that the angle BDC and half the angle BAC make up a right
angle.

08. If the exterior angle of a triangle be bisected, and the angles
of the triangle made by the bisectors be bisected, and so on, the
triangles so formed will tend to become eventually equilateral.

59. If in the three sides AB, BC, CA of an equilateral triangle
ABC, distances AF, BF, CG be taken, each equal to a third of
one of the sides, and the points F, F, G be respectively joined
(1) with each other, (2) with the opposite angles : shew that the two
triangles so formed, are equilateral triangles.

IV.

60. Describe a right-angled triangle upon a given base, having
given also the perpendicular from the right angle upon the hy-
potenuse.

61. Given one side of a right-angled triangle, and the difference
between the hypotenuse and the sum of the other two sides, to con-
struct the triangle.

62. Construct an isosceles right-angled triangle, having given
(1) the sum of the hypotenuse and one side ; (2) their difference.

63. Describe a right-angled triangle of which the hypotenuse
and the difference between the other two sides are given.

64. Given the base of an isosceles triangle, and the sum or dif-
ference of a side and the perpendicular from the vertex on the base.
Construct the triangle.

65. Make an isosceles triangle of given altitude whose sides shall
pass through two given points and have its base on a given straight
line.

66. Construct an equilateral triangle, having given the length of
the perpendicular drawn from one of the angles on the opposite side.

67. Having given the straight lines which bisect the angles at the
base of an equilateral triangle, determine a side of the triangle.

68. Having given two sides and an angle of a triangle, construct
the triangle, distinguishing the different cases.

69. Having given the base of a triangle, the difference of the sides,
and the difference of the angles at the base ; to describe the triangle.

70. Given the perimeter and the angles of a triangle, to con-
struct it.

71. Having given the base of a triangle, and half the sum and
half the difference of the angles at the base ; to construct the triangle.

72. Having given two lines, which are not parallel, and a point
between them; describe a triangle having two of its angles in the
respective lines, and the third at the given point ; and such that the
sides shall be equally inclined to the lines which they meet.

73. Construct a triangle, having given the three lines drawn from
the angles to bisect the sides opposite.

ON BOOK I. 79

74. Given one of the angles at the base of a triangle, the base
itself, and the sum of the two remaining sides, to construct the tri-
angle.

75. Given the base, an angle adjacent to the base, and the dif-
ference of the sides of a triangle, to construct it.

76. Given one angle, a side opposite to it, and the difference of
the other two sides ; to construct the triangle.

77. Given the base and the sum of the two other sides of a
triangle, construct it so that the line which bisects the vertical
angle shall be parallel to a given line.

V.

78. Prom a given point without a given straight line, to draw a line
making an angle with the given line equal to a given rectilineal angle.

79. Through a given point A, draw a straight line ^5C meeting
two given parallel straight lines in B and C, such that BC may be
equal to a given straight line.

80. If the line joining two parallel lines be bisected, all the lines
di'awn through the point of bisection and terminated by the parallel
lines are also bisected in that point.

81. Three given straight lines issue from a point: draw another
straight line cutting them so that the two segments of it intercepted
between them may be equal to one another.

82. AB, AC are two straight lines, B and C given points in the
same; BD is drawn perpendicular to AC, and DE perpendicular to
AB; in like manner Ci'' is drawn perpendicular to AB, and FG to
A C. Shew that JEG is parallel to B C.

83. ABC is a right-angled triangle, and the sides AC, AB are
produced to D and F; bisect FBC and BCD by the lines BF, CF,
and from F let fall the perpendiculars FF, ED. Prove (without
assuming any properties of parallels) that ADFF is a square.

84. Two pairs of equal straight lines being given, shew how to
construct with them the greatest parallelogram.

85. With two given lines as diagonals describe a parallelogram
which shall have an angle equal to a given angle. Within what
limits must the given ang^e lie ?

86. Having given one of the diagonails of a parallelogram, the
sum of the two adjacent sides and the angle between them, construct
the parallelogram.

87. One of the diagonals of a parallelogram being given, and the
angle which it makes with one of the sides, complete the parallelo-
gram, so that the other diagonal may be parallel to a given line.

88. A BCD, A' BCD' are two parallelograms whose corres-
ponding sides are equal, but the angle A is greater than the angle
A', prove that the diameter A C is less than A' C, but BD greater
than B'D\

89. If in the diagonal of a parallelogram any two points equi-
distant from its extremities be joined with the opposite angles, a
figure will be formed w^hich is also a parallelogram.

90. From each angle of a parallelogram a line is drawn making

80 GEOMETRICAL EXERCISES

the same angle towards the same parts with an adjacent side, taken
always in the same order ; shew that these lines form another parallelo-
gram similar to the original one.

91. Along the sides of a parallelogram taken in order, measure
AA' = BB' = CC' = DD' : the figure A' BCD' will be a parallelogram.

92. On the sides AB, BC, CD, DA, of a parallelogram, set off
AE, BF, CG, DII, equal to each other, and join AF, BG, CH,DE:
these lines form a j)arallelogram, and the difference of the angles
AFB, BGC, equals the difference of any two proximate angles of the
two parallelograms.

93. OB, OC are two straight lines at right angles to each other,
through any point P any two straight lines are drawn intersecting
OB, OC, in B, B', C, C, respectively. If D and D be the middle
points of BB and CO^ shew that the angle B PD' is equal to the
angle DOD.

94. A BCD is a parallelogram of which the angle Cis opposite to
the angle A. If through A any straight line be drawn, then the dis-
tance of C is equal to the sum or difference of the distances of B and
of D from that straight line, according as it lies without or within the
parallelogram.

95. tjpon stretching two chains AC, BD, across a field ABCD,
I find that ^Z) and ^ C make equal angles with DC, and that AC
makes the same angle with AD that BD does with BC-, hence prove
that AB is parallel to CD.

96. To find a point in the side or side produced of any parallelo-
gram, such that the angle it makes with the line joining the point
and one extremity of the opposite side, may be bisected by the line
joining it with the other extremity.

97. When the corner of the leaf of a book is turned down a second
time, so that the lines of folding are parallel and equidistant, the space
in the second fold is equal to three times that in the first.

VI.

98. If the points of bisection of the sides of a triangle be joined,
the triangle so formed shall be one-fourth of the given triangle.

99. If in the triangle ABC, BC be bisected in D, AD joined
and bisected in F, BF joined and bisected in F, and CF joined and
bisected in G ; then the triangle FFG will be equal to one-eighth of
the triangle ABC.

100. Shew that the areas of the two equilateral triangles in
Prob. 59, p. 78, are respectively, one- third and one-seventh of the area
of the original triangle.

101. To describe a triangle equal to a given triangle, (1) when
the base, (2) when the altitude of tlie required triangle is given.

102. To describe a triangle equal to the sum or difference of two
given triangles.

103. Upon a given base describe an isosceles triangle equal to a
given triangle.

104. Describe a right-angled triangle equal to a given triangle
ABC,

205, To a given straight line apply a triangle which shall be equal

ON BOOK I." 81

to a given parallelogram and have one of its angles equal to a given
rectilineal angle.

106. Transform a given rectilineal figure into a triangle whose
vertex shall be in a given angle of the figure, and whose base shall be
in one of the sides.

107. Divide a triangle by two straight lines into three parts which
when properly arranged shall form a parallelogram whose angles are
of a given magnitude.

108. Shew that a scalene triangle cannot be divided into two
parts which will coincide.

109. If two sides of a triangle be given, the triangle will be
greatest when they contain a right angle.

110. Of all triangles having the same vertical angle, and whose
bases pass through a given point, the least is that whose base is bisected
in the given point.

111. Of all triangles having the same base and the same perimeter,
that is the greatest which has the two undetermined sides equal.

112. Divide a triangle into three equal parts, (1) by lines drawn
from a point in one of the sides : (2) by lines drawn from the angles
to a point within the triangle : (3) by lines di-awn from a given point
within the triangle. In how many ways can the third case be done ?

113. Divide an equilateral triangle into nine equal parts.

114. Bisect a parallelogram, (1) by a line drawn from a point in
one of its sides : (2) by a line drawn from a given point within or
without it : (3) by a line perpendicular to one of the sides : (4) by a
line drawn parallel to a given line.

115. From a given point in one side produced of a parallelogram,
draw a straight line which shall divide the parallelogram into two
equal parts.

116. To trisect a parallelogram by lines drawn (1) from a given
point in one of its sides, (2) from one of its angular points.

VII.

117. To describe a rhombus which shall be equal to any given

118. Describe a parallelogram which shall be equal in area and
perimeter to a given triangle.

119. Find a point in the diagonal of a square produced, from which
if a straight line be drawn parallel to any side of the square, and
meeting another side produced, it will form together with the pro-
duced diagonal and produced side, a triangle equal to the square.

120. If from any point within a parallelogram, straight lines be
drawn to the angles, the parallelogram shall be divided into four tri-
angles, of which each two opposite are together equal to one-half of
the parallelogram.

121. If AB CD be a parallelogram, and B any point in the dia-
gonal A C, or ^ C produced ; shew that the triangles JEBC, EDO, are
equal, as also the triangles EBA and EBD.

122. ABCD is a parallelogram, draw DFG meeting BC in F,

e5

â–

82

GEOMETRICAL EXERCISES

and AB produced in G ; join AF, CG ; then will the triangles ABF,
CFG be equal to one another.

123. aBCD is a parallelogram, B the point of intersection of its
diagonals, and K any point in AB. If KB, KC be joined, shew that
the figure BKECh one-fourth of the parallelogram.

124. Let ABCD be a parallelogram, and O any point within it,
through O draw lines parallel to the sides of ABCD, and join OA,
OC', prove that the difference of the parallelograms DO, BO is twice
the triangle OA C.

125. The diagonals A C, BD of a parallelogram intersect in 0, and
P is a point within the triangle ^OJ? ; prove that the difference of the
triangles APB, CPD is equal to the sum of the triangles APC, BPD.

11:6. UK be the common angular point of the parallelograms
about the diameter -4C(fig. Euc. I. 43.) and BD be the other dia-
meter, the difference of these parallelograms is equal to twice the
triangle BKD.

127. The perimeter of a square is less than that of any other paral-
lelogram of equal area.

128. Shew that of all equiangular parallelograms of equal peri-
meters, that which is equilateral is the greatest.

129. Prove that the perimeter of an isosceles triangle is greater
than that of an equal right-angled parallelogram of the same altitude.

VIIL

130. If a quadrilateral figure is bisected by one diagonal, the
second diagonal is bisected by the first.

131. If two opposite angles of a quadrilateral figure are equal,
shew that the angles between opposite sides produced are equal.

132. Prove that the sides of any four-sided rectilinear figure are
together greater than the two diagonals. ^

133. The sum of the diagonals of a trapezium is less than the sum<
of any four lines which can be drawn to the four angles, from any
point within the figure, except their intersection.

134. The longest side of a given quadrilateral is opposite to the
shortest : shew that the angles adjacent to the shortest side are together
greater than the sum of the angles adjacent to the longest side.

135. Give any two points in the opposite sides of a trapezium, in-
scribe in it a parallelogram having two of its angles at these points.

136. Shew that in every quadrilateral plane figure, two parallelo-
grams can be described upon two opposite sides as diagonals, such
that the other two diagonals shall be in the same straight line and equal.

137. Describe a quadrilateral figure whose sides shall be equal to
four given straight lines. What limitation is necessary ?

138. If the sides of a quadrilateral figure be bisected and the
points of bisection joined, the included figure is a parallelogram, and
equal in area to half the original figure.

139. A trapezium is such, that the perpendiculars let fall on a
diagonal from the opposite angles are equal. Divide the trapezium
into four equal triangles, by straight lines drawn to the angles from
TDoint within it.

ON BOOK 1. 86

140. If two opposite sides of a trapezium be parallel to one another,
the straight line joining their bisections, bisects the trapezium.

141. If of the four triangles into which the diagonals divide a
trapezium, any two opposite ones are equal, the trapezium has two of
its opposite sides parallel.

142. If two sides of a quadrilateral are parallel but not equal,
and the other two sides are equal but not parallel, the opposite angles
of the quadrilateral are together equal to two right angles: and
conversely.

143. If two sides of a quadrilateral be parallel, and the line joining
the middle points of the diagonals be produced to meet the other
sides ; the line so produced will be equal to half the sum of the
parallel sides, and the line between the points of bisection equal to
half their difference.

144. To bisect a trapezium, (1) by a line drawn from one of its
angular points : (2) by a line drawn from a given point in one side.

145. To divide a square into four equal portions by lines drawn
from any point in one of its sides.

146. It is impossible to divide a quadrilateral figure (except it be
a parallelogram) into equal triangles by lines drawn from a point
within it to its four corners.

IX.

147. If the greater of the acute angles of a right-angled triangle,
be double the other, the square on the greater side is three times the
square on the other.

148. Upon a given straight line construct a right-angled triangle
such that the square on the other side may be equal to seven times
the square on the given line.

149. If from the vertex of a plane triangle, a perpendicular fall
upon the base or the base produced, the dift'erence of the squares on
the sides is equal to the difference of the squares on the segments of
the base.

150. If from the middle point of one of the sides of a right-angled
triangle, a perpendicular be drawn to the hypotenuse, the difference
of the squares on the segments into which it is divided, is equal to the
square on the other side.

151. If a straight line be drawn from one of the acute angles of a
right-angled triangle, bisecting the opposite side, the square upon that
line is less than the square upon the hypotenuse by three ' times the
square upon half the line bisected.

152. If the sum of the squares on the three sides of a triangle be
equal to eight times the square on the line drawn from the vertex
to the point of bisection of the base, then the vertical angle is a
right angle.

153. If a line be drawn parallel to the hypotenuse of a right-
angled triangle, and each of the acute angles be joined with the
points where this line intersects the sides respectively opposite to
them, the squares on the joining lines are together equal to the
squares on the hypotenuse and on the line drawn parallel to it.

i

84 GEOMETRICAL EXERCISES ON BOOK I.

154. Let ACB, ADB be two right-angled triangles having a
common hypotenuse AB, join CD, and on CD produced both ways
draw perpendiculars AE, BF. Shew that CE^ + CF' = DE^ + DF\

155. If perpendiculars ^ J), BE, CF drawn from the angles on
the opposite sides of a triangle intersect in G, the squares on AB^
BC, and CA, are together three times the squares on AG, BG.
and CG,

156. If ABC be a triangle of which the angle ^ is a right
angle; and BE, CF be drawn bisecting the opposite sides re-
spectively: shew that four times the sum of the squares on BE
and Ci^ is equal to five times the square on BC.

157. If ABC be an isosceles triangle, and CD be drawn per-
pendicular to AB; the sum of the squares on the three sides is
equal to

158. The sum of the squares described upon the sides of a rhombus
is equal to the squares described on its diameters.

159. A point is taken within a square, and straight lines drawn
from it to the angular points of the square, and perpendicular to the
sides ; the squares on the first are double the sum of the squares on
the last. Shew that these sums are least when the point is in the
center of the square.

160. In the figure Euc. I. 47,

(a) Shew that the diagonals FA, AK of the squares on AB, A C,
lie in the same straight line.

(b) If DF, EKhe joined, the sum of the angles sit the bases
of the triangles BFD, CEK is equal to one right angle.

(c) If BG and CShe joined, those lines will be parallel.

{d) If perpendiculars be let fall from F and K on BC produced,
the parts produced will be equal; and the perpendiculars together
will be equal to B C.

(e) Join GH, KE, FD, and prove that each of the triangles so