Geometrical Problems Deducible from the First Six Books of Euclid: Arranged and Solved: to which is Added an Appendix Containing the Elements of Plane Trigonometry ...

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J. Smith, 1827 - Euclid's Elements - 377 pages
 

Contents

Given the vertical angle the radius of the inscribed circle
28
If on the radius of a given semicircle another semicircle
30
points of intersection to the extremities of the diameter cutting each
36
of intersection a circle be described cutting them the points where
42
If from the angular points of the squares described upon
43
cumferences the lines joining the points of intersection and
47
other extremity of the diameter the part without the circle may
52
being in the circumference of the other and any line be drawn from
58
line which shall make with the circumference an angle less than
63
If the diagonals of a quadrilateral figure inscribed in a circle
65
be drawn perpendicular to the base and from the greater segment
69
to meet the tangents drawn from the extremities of the bisecting line
75
from the points of intersection and produced to the circumference
80
intersecting circles and a line drawn from one extremity of
84
If two circles cut each other and any two points be taken
87
From a given point in the diameter of a semicircle produced
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the other two sides
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drawn to the opposite sides making equal angles with the base
97
If two exterior angles of a triangle be bisected and from
103
The three straight lines which bisect the three angles
109
secting that which joins the vertex and the bisection of the base
114
If the opposite sides or opposite angles of a quadrilateral
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of any four lines which can be drawn to the four angles from
126
If the sides of an equilateral and equiangular hexagon be pro
135
drawn to the angular points the sums of the squares of those which
140
To describe a rectangular parallelogram which shall be equal
185
If the opposite angles of a quadrilateral figure be equal
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a triangle to describe on the other sides segments similar to that
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In a given triangle to inscribe a parallelogram similar
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points and touch a given straight line
203
and touch a given circle and a given straight line
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line both given in position and have its centre also in a given
210
To draw two lines parallel to the adjacent sides of a given
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be drawn to cut one another the greater segments will be equal
220
cut each other at right angles the rectangles contained by the oppo
226
base tangents be drawn intersecting their circumferences the points
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If a triangle be inscribed in a circle and from its vertex
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In any triangle if perpendiculars be drawn from the angles
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drawn through the centre of its inscribed circle and a perpendicular
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If the exterior angle of a triangle be bisected by a straight
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If two points be taken in the diameter of a circle equidistant
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tained by the segments of the diameter will be less or greater than
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be drawn to any point in the circumference meeting a diameter per
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drawn to any point in the circumference and meeting the perpen
277
described with radii equal the former to the side and the latter
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the point of contact another be described with the same radius
289
Given the vertical angle the line bisecting the base and
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taining it and the difference of the segments of the base made
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and the rectangle contained by the straight lines drawn from
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bisecting the vertical angle
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Page 12 - IF a straight line be divided into two equal, and also into two unequal parts ; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section.
Page xv - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle,. shall be equal to the square of the line which touches it.
Page xxx - AB be the given straight line ; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.
Page 303 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees ; each degree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds.
Page 140 - Iff a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other...
Page 329 - CE is equal to the difference of the segments of the base made by the perpendicular.
Page 109 - If from a point, without a parallelogram, there be drawn two straight lines to the extremities of the two opposite sides, between which, when produced, the point does not lie, the difference of the triangles thus formed is equal to half the parallelogram. Ex. 2. The two triangles, formed by drawing straight lines from any point within a parallelogram to the extremities of its opposite sides, are together half of the parallelogram.
Page 164 - PROPOSITION I. PROBLEM. — To describe an equilateral triangle upon a given finite straight line. Let AB be the given straight line; it is required to describe an equilateral triangle upon it.
Page 281 - Given the vertical angle, the difference of the two sides containing it, and the difference of the segments of the base made by a perpendicular from the vertex ; construct the triangle.
Page 270 - AB describe a segment of a circle containing an angle equal to the given angle, (in.

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