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, 1819 - 377 Seiten
 

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Two straight lines and a point in each are given in position
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circle
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If from the extremities of the diameter of a semicircle per
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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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triangles equiangular and equal to the given triangle
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pendiculars to their common diameter be produced to cut the cir
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the given point may together be equal to a given line not greater
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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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the circle
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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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triangle meet in the same point
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adjacent to the angles is equal to the square of either line drawn
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To draw a line from one of the angles at the base of a tri
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of its sides
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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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hypothenuse produced they will cut off equal segments and
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mine a third such that its distances from the extremities may
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contained by the whole and one of the parts may be equal to
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to a given square and have the difference of its adjacent sides equal
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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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given parallelogram
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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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line both given in position and have its centre also in a given
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point
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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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drawn parallel to the other intersecting the adjacent side of
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circle be produced till they meet the three points of intersection
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with a radius less than one of the equal sides but greater than
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If a triangle be inscribed in a semicircle and a perpen
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described on the sides of a rightangled triangle is in the middle
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Those segments are also in the duplicate ratio of
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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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If a diameter of a circle be produced to bisect a line at right
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If from any point in the base or base produced of the seg
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If from the point of bisection and any other point in a given
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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
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by the perpendicular the sum of the squares of the sides and
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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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Given the base one of the angles at the base and the differ
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Seite 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.
Seite 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.
Seite 230 - To describe an isosceles triangle, having each of the angles at the base double of the third angle.
Seite 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.
Seite 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...
Seite 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.
Seite 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.
Seite 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.
Seite 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.
Seite 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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