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 - Geometry - 377 pages
 

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Contents

If from two given points straight lines be drawn containing
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Page
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angles
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If any number of equal straight lines be placed in a circle
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points of intersection to the extremities of the diameter cutting each
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of intersection a circle be described cutting them the points where
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other extremity of the diameter the part without the circle may
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the other
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the given point may together be equal to a given line not greater
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the circle
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being in the circumference of the other and any line be drawn from
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greater may be double the part intercepted by the circumference
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line which shall make with the circumference an angle less than
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be drawn perpendicular to the base and from the greater segment
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In a given circle to draw a chord parallel to a straight line
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to meet the tangents drawn from the extremities of the bisecting line
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from the points of intersection and produced to the circumference
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the whole line and one of the parts as diameters semicircles
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From a given point in the diameter of a semicircle produced
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the other two sides
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the base
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In a given circle to place a straight line cutting two radii
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drawn to the opposite sides making equal angles with the base
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the point of intersection of the bisecting lines a line be drawn to
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In a given triangle to draw a line parallel to one of
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by this line produced and one of the others is equal to the angle
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If from either angle of a triangle a line be drawn inter
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If the opposite sides or opposite angles of a quadrilateral
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If a straight line be drawn parallel to one of the sides
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bisection of the sides of a trapezium and its ratio to the trapezium
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If the sides of an equilateral and equiangular hexagon be pro
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al which shall be a mean proportional between the segments of the side
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nated by two others given in position so as to form with them
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Between two lines given in position to draw a line equal
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Page
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to a given square and have its adjacent sides together equal
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two right angles a circle may be described about
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If an equilateral triangle be inscribed in a circle the square
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In a given triangle to inscribe a triangle similar to a given
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points and touch a given straight line
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and touch a given circle and a given straight line
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To describe a circle which shall touch a circle and straight
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figure equal to the triangle and having two of its sides parallel
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be drawn to cut one another the greater segments will be equal
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gon drawn parallel to the other intersecting the adjacent side of
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If the opposite sides of an irregular hexagon inscribed
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described on the sides of a rightangled triangle is in the middle
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be centre of the inner and produced to the circumference of the outer
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through the points where they meet that line and the point in
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drawn to any point in the circumference and meeting the perpen
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If the exterior angle of a triangle be bisected by a straight
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from the centre the sum of the squares of the two lines drawn from
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tained by the segments of the diameter will be less or greater than
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out pendicular to the chord the rectangle contained by the distances
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arc of a circle two parallel lines be drawn the former terminated
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described with radii equal the former to the side and the latter
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If a straight line be divided into any two parts and upon
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Given one angle a side opposite to it and the sum of
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at the base to construct the triangle
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taining it and the difference of the segments of the base made
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contact
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of the diameter of a semicircle from which if a line be drawn to
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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 305 - 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 142 - 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 331 - CE is equal to the difference of the segments of the base made by the perpendicular.
Page 111 - 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 166 - 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 283 - 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 272 - AB describe a segment of a circle containing an angle equal to the given angle, (in.

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