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 - Euclid's Elements - 377 pages

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points of intersection to the extremities of the diameter cutting each
of intersection a circle be described cutting them the points where
If from the angular points of the squares described upon
pendiculars to their common diameter be produced to cut the cir
other extremity of the diameter the part without the circle may
being in the circumference of the other and any line be drawn from
line which shall make with the circumference an angle less than
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
If two circles cut each other and any two points be taken
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
The three straight lines which bisect the three angles of
To draw a line from one of the angles at the base of a tri
If the opposite sides or opposite angles of a quadrilateral
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
sides of a rightangled triangle perpendiculars be let fall upon
point in it which is equally distant from the upper end of the line
circumferences so that the rectangle contained by the two chords
triangle to describe on the other sides segments similar to that
In a given triangle to inscribe a parallelogram similar to
through a given point and touch a circle in a given point the
line both given in position and have its centre also in a given
On the base of a given triangle to describe a quadrilateral
be drawn to cut one another the greater segments will be equal
drawn parallel to the other intersecting the adjacent side of
If the opposite sides of an irregular hexagon inscribed in
a circle is greater or less than a right angle by the angle contained
from its extremity the line intercepted between the vertex and
base tangents be drawn intersecting their circumferences the points
If a triangle be inscribed in a circle and from its vertex
In any triangle if perpendiculars be drawn from the angles
drawn through the centre of its inscribed circle and a perpendicular
If the exterior angle of a triangle be bisected by a straight
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
If two equal circles touch each other externally and through
other two sides to construct the triangle
taining it and the difference of the segments of the base made
bisecting the vertical angle
by the perpendicular the sum of the squares of the sides and

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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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