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EXPERIMENTAL PROOFS OF PYTHAGORAS'S THEOREM
THEOREM 30. [Euc. I. 48.] If the square described on one side
of a triangle is equal to the sum of the squares described on
the other two sides, then the angle contained by these two
sides is a right angle.
PROBLEM 16. To draw squares whose areas shall be respectively
twice, three-times, four-times, that of a given square.
Problems on Areas.
PROBLEM 17. To describe a parallelogram equal to a given
triangle, and having one of its angles equal to a given angle.
PROBLEM 18. To draw a triangle equal in area to a given
quadrilateral.
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122
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126
128
PROBLEM 19. To draw a parallelogram equal in area to a given
rectilineal figure, and having an angle equal to a given angle. 129
SYMMETRICAL PROPERTIES OF CIRCLES
139
141
THEOREM 31. [Euc. III. 3.] If a straight line drawn from
the centre of a circle bisects a chord which does not pass
through the centre, it cuts the chord at right angles.
Conversely, if it cuts the chord at right angles, it bisects it.
COR. 1. The straight line which bisects a chord at right
angles passes through the centre.
COR. 2. A straight line cannot meet a circle at more than
two points.
COR. 3. A chord of a circle lies wholly within it.
THEOREM 32. One circle, and only one, can pass through any
three points not in the same straight line.
144
145.
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146
COR. 1. The size and position of a circle are fully deter-
mined if it is known to pass through three given points.
147
COR. 2. Two circles cannot cut one another in more than
two points without coinciding entirely.
HYPOTHETICAL CONSTRUCTION
THEOREM 33. [Euc. III. 9.] If from a point within a circle
more than two equal straight lines can be drawn to the
circumference, that point is the centre of the circle.
THEOREM 34. [Euc. III. 14.] Equal chords of a circle are equi-
distant from the centre.
Conversely, chords which are equidistant from the centre
are equal.
THEOREM 35. [Euc. III. 15.] Of any two chords of a circle,
that which is nearer to the centre is greater than one more
remote.
Conversely, the greater of two chords is nearer to the
centre than the less.
COR. The greatest chord in a circle is a diameter.
THEOREM 36. [Euc. III. 7.] If from any internal point, not
the centre, straight lines are drawn to the circumference of a
circle, then the greatest is that which passes through the
centre, and the least is the remaining part of that diameter.
And of any other two such lines the greater is that which
subtends the greater angle at the centre.
THEOREM 37. [Euc. III. 8.] If from any external point
straight lines are drawn to the circumference of a circle, the
greatest is that which passes through the centre, and the
least is that which when produced passes through the centre.
And of any other two such lines, the greater is that which
Angles in a Circle.
THEOREM 38. [Euc. III. 20.] The angle at the centre of a
circle is double of an angle at the circumference standing on
the same arc.
THEOREM 39. [Euc. III. 21.] Angles in the same segment of a
circle are equal.
CONVERSE OF THEOREM 39. Equal angles standing on the
same base, and on the same side of it, have their vertices on
an arc of a circle, of which the given base is the chord.
THEOREM 40. [Euc. III. 22.] The opposite angles of any
quadrilateral inscribed in a circle are together equal to two
right angles.
CONVERSE OF THEOREM 40. If a pair of opposite angles
of a quadrilateral are supplementary, its vertices are con-
cyclic.
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THEOREM 41. [Euc. III. 31.] The angle in a semi-circle is a
right angle.
164
COR. The angle in a segment greater than a semi-circle is
acute; and the angle in a segment less than a semi-circle is
obtuse.
165
THEOREM 42. [Euc. III. 26.] In equal circles, arcs which sub-
tend equal angles, either at the centres or at the circum-
ferences, are equal.
COR. In equal circles sectors which have equal angles are
equal.
THEOREM 43. [Euc. III. 27.] In equal circles angles, either at
the centres or at the circumferences, which stand on equal
arcs are equal.
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167
THEOREM 44. [Euc. III. 28.] In equal circles, arcs which are
cut off by equal chords are equal, the major arc equal to the
major arc, and the minor to the ninor.
168
THEOREM 45. [Euc. III. 29.] In equal circles chords which
cut off equal arcs are equal.
169
Tangency.
DEFINITIONS AND FIRST PRINCIPLES -
172
THEOREM 46. The tangent at any point of a circle is perpendi-
cular to the radius drawn to the point of contact./
174
COR. 1. One and only one tangent can be drawn to a
circle at a given point on the circumference.
COR. 2. The perpendicular to a tangent at its point of
contact passes through the centre.
COR. 3. The radius drawn perpendicular to the tangent
passes through the point of contact.
THEOREM 47. Two tangents can be drawn to a circle from an
external point.
176
COR. The two tangents to a circle from an external point
are equal, and subtend equal angles at the centre.
THEOREM 48. If two circles touch one another, the centres and
the point of contact are in one straight line.
178
COR. 1. If two circles touch externally the distance be-
tween their centres is equal to the sum of their radii.
COR. 2. If two circles touch internally, the distance be-
tween their centres is equal to the difference of their radii.
THEOREM 49. [Euc. III. 32.] The angles made by a tangent
to a circle with a chord drawn from the point of contact are
respectively equal to the angles in the alternate segments of
the circle.
180
Problems.
GEOMETRICAL ANALYSIS
182
PROBLEM 20. Given a circle, or an arc of a circle, to find its
centre.
183
PROBLEM 21. To bisect a given arc.
PROBLEM 22.
To draw a tangent to a circle from a given ex-
ternal point.
PROBLEM 23. To draw a common tangent to two circles.
THE CONSTRUCTION OF CIRCLES -
PROBLEM 24. On a given straight line to describe a segment of
a circle which shall contain an angle equal to a given angle.
COR. To cut off from a given circle a segment containing
a given angle, it is enough to draw a tangent to the circle,
and from the point of contact to draw a chord making with
the tangent an angle equal to the given angle.
Circles in Relation to Rectilineal Figures.
DEFINITIONS
PROBLEM 25. To circumscribe a circle about a given triangle.
PROBLEM 26. To inscribe a circle in a given triangle.
PROBLEM 27. To draw an escribed circle of a given triangle.
PROBLEM 28. In a given circle to inscribe a triangle equi-
angular to a given triangle.
PROBLEM 29. About a given circle to circumscribe a triangle
equiangular to a given triangle.
197
PROBLEM 30. To draw a regular polygon (i) in (ii) about a
given circle.
200
PROBLEM 31. To draw a circle (i) in (ii) about a regular polygon. 201
THEOREM 50. [Euc. II. 1.] If of two straight lines, one is
divided into any number of parts, the rectangle contained by
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the two lines is equal to the sum of the rectangles con-
tained by the undivided line and the several parts of the
divided line.
COROLLARIES.
[Euc. II. 2 and 3.]
THEOREM 51. [Euc. II. 4.] If a straight line is divided in-
ternally at any point, the square on the given line is equal to
the sum of the squares on the two segments together with
twice the rectangle contained by the segments.
THEOREM 52. [Euc. II. 7.] If a straight line is divided
externally at any point, the square on the given line is equal
to the sum of the squares on the two segments diminished by
THEOREM 53. [Euc. II. 5 and 6.] The difference of the squares
on two straight lines is equal to the rectangle contained by
their sum and difference.
COR. If a straight line is bisected, and also divided (inter-
nally or externally) into two unequal segments, the rectangle
contained by these segments is equal to the difference of the
squares on half the line and on the line between the points of
section.
THEOREM 54. [Euc. II. 12.] In an obtuse-angled triangle, the
square on the side subtending the obtuse angle is equal to the
sum of the squares on the sides containing the obtuse angle
together with twice the rectangle contained by one of those
sides and the projection of the other side upon it..
THEOREM 55. [Euc. II. 13.] In every triangle the square on
the side subtending an acute angle is equal to the sum of the
squares on the sides containing that angle diminished by
twice the rectangle contained by one of those sides and the
projection of the other side upon it.
THEOREM 56. In any triangle the sum of the squares on two
sides is equal to twice the square on half the third side
together with twice the square on the median which bisects
the third side.
Rectangles in connection with Circles.
THEOREM 57. [Euc. III. 35.] If two chords of a circle cut at a
point within it, the rectangles contained by their segments
THEOREM 58. [Euc. III. 36.] If two chords of a circle, when
produced, cut at a point outside it, the rectangles contained
by their segments are equal. And each rectangle is equal to
the square on the tangent from the point of intersection.
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