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

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If any number of equal straight lines be placed in a circle
points of intersection to the extremities of the diameter cutting each
of intersection a circle be described cutting them the points where
triangles equiangular and equal to the given triangle
pendiculars to their common diameter be produced to cut the cir
equal to a given straight line
being in the circumference of the other and any line be drawn from
To determine a point in the arc of a quadrant from which
be drawn perpendicular to the base and from the greater segment
to meet the tangents drawn from the extremities of the bisecting line
in the circumference of one of them through which lines are drawn
which are perpendicular to each other in such a manner that
the whole line and one of the parts as diameters semicircles
From a given point in the diameter of a semicircle produced
the other two sides
drawn to the opposite sides making equal angles with the base
If two exterior angles of a triangle be bisected and from
triangle meet in the same point
To draw a line from one of the angles at the base of a tri
figure be equal the figure will be a parallelogram
of any four lines which can be drawn to the four angles from
If the sides of an equilateral and equiangular pentagon
perpendiculars be let fall on every side the sum of the squares
triangle and the extremities of the adjacent sides be joined
hypothenuse produced they will cut off equal segments and
centre at right angles to the plane of a circle to determine that
two right angles a circle may be described about
triangle to describe on the other sides segments similar to that
given parallelogram
points and touch a given straight line
and touch a given circle and a given straight line
line both given in position and have its centre also in a given
figure equal to the triangle and having two of its sides parallel
be drawn to cut one another the greater segments will be equal
drawn parallel to the other intersecting the adjacent side of
circle be produced till they meet the three points of intersection
a circle is greater or less than a right angle by the angle contained
from its extremity the line intercepted between the vertex and
described on the sides of a rightangled triangle is in the middle
adjacent to the angles is equal to the square of either line drawn
In any triangle if perpendiculars be drawn from the angles
drawn through the centre of its inscribed circle and a perpendicular
line which cuts the base produced the square of the bisecting line
the chord of an arc the square of that line together with the rectangle
If a diameter of a circle be produced to bisect a line at right
be drawn to any point in the circumference meeting a diameter per
drawn to any point in the circumference and meeting the perpen
described with radii equal the former to the side and the latter
the point of contact another be described with the same radius
other two sides to construct the triangle
bisecting the vertical angle
by the perpendicular the sum of the squares of the sides and

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Strona 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.
Strona 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.
Strona 230 - To describe an isosceles triangle, having each of the angles at the base double of the third angle.
Strona 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.
Strona 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...
Strona 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.
Strona 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.
Strona 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.
Strona 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.
Strona 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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