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 ... (Google eBook)

Front Cover
Printed by J. Smith, 1819 - Geometry - 377 pages
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Contents

If from a point without a circle two straight lines be drawn
25
of chords be drawn the locus of their points of bisection will be
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points of intersection to the extremities of the diameter cutting each
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of intersection a circle1 be described cutting them the points where
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If from the angular points of the squares described upon
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pendiculars to their common diameter be produced to cut the cir
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other extremity of the diameter the part without the circle may
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being in the circumference of the other and any line be drawn from
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To determine a point in the arc of a quadrant from which
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be drawn perpendicular to the base and from the greater segment
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to meet the tangents drawn from the extremities of the bisecting line
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in the circumference of one of them through which lines are drawn
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which are perpendicular to each other in such a manner that
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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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drawn to the opposite sides making equal angles with the base
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If two exterior angles of a triangle be bisected and from
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The three straight lines which bisect the three angles of
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To draw a line from one of the angles at the base of a tri
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To bisect a parallelogram by a line drawn from a point in
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of any four lines which can be drawn to the four angles from
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If the sides of an equilateral and equiangular pentagon
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perpendiculars be let fall on every side the sum of the squares
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triangle and the extremities of the adjacent sides be joined
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Between two lines given in position to draw a line equal
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mainders may have a given ratio and the sum of the squares of
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to a given square and have its adjacent sides together equal to
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two right angles a circle may be described about
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triangle to describe on the other sides segments similar to that
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In a given triangle to inscribe a parallelogram similar to
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To describe a circle the centre of which may be in the per
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To describe a circle which shall touch a straight line
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line both given in position and have its centre also in a given
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On the base of a given triangle to describe a quadrilateral
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be drawn to cut one another the greater segments will be equal
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a circle is greater or less than a right angle by the angle contained
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from its extremity the line intercepted between the vertex and
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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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If a diameter of a circle be produced to bisect a line at right
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drawn to any point in the circumference and meeting the perpen
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described with radii equal the former to the side and the latter
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If two equal circles touch each other externally and through
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other two sides to construct the triangle
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taining it and the difference of the segments of the base made
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bisecting the vertical angle
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by the perpendicular the sum of the squares of the sides and
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Common terms and phrases

Popular passages

Page 14 - 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 xiii - 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 230 - To describe an isosceles triangle, having each of the angles at the base double of the third angle.
Page 327 - 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 158 - 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 212 - FC are equal to one another : wherefore the circle described from the centre F, at the distance of one of them, will pass through the extremities of the other two, and be described about the triangle ABC.
Page 123 - 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 305 - 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 247 - The perpendicular from the vertex on the base of an equilateral triangle is equal to the side of an equilateral triangle inscribed in a circle, whose diameter is the base.
Page 299 - AB be equal to the given bisecting line ; and upon it describe a segment of a circle containing an angle equal to the given angle.

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