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104. From the centre of a given circle draw a straight line to meet a given tangent to the circle, so that the segment of the line between the circle and the tangent shall be any required part of the tangent.
105. Given in any triangle the base, the ratio of the sides, and the distance between the points, in which the internal and external bisectors cut the base, construct the triangle.
106. AB is the diameter of a circle, D any point in the circumference, and C the middle point of the arc AD. If AC, AD, BC, be joined, and AD cut BC in E, the circle described about the triangle AEB will touch AC, and its diameter will be a third proportional to BC and AB.
107. From a given point ▲ a variable straight line is drawn, meeting a fixed straight line on P, and a point Q is taken on it so that the rectangle AP, AQ is constant. Find the locus of Q.
108. On a given base describe a rectangle, which shall be equal to the difference of the squares on two given straight lines, any two of the three given lines being together greater than the third.
109. If the exterior angles of a triangle be bisected by straight lines, forming another triangle, shew that the two triangles cannot be similar, unless they be each equilateral.
110. If ABC, A'B'C' be similar triangles, and AB=A'C', shew the areas of the triangles are as AC to A'B'.
111. The alternate angles of a regular hexagon are joined: shew that the area of the hexagon formed by the intersections of the joining lines is one-third of the original hexagon.
112. A triangle is divided by a straight line parallel to the base into two parts, the areas of which are as 1 to 8: how does the straight line divide the sides ?
113. The line AD is divided into three equal parts in the points B and C; a circle is described with B as centre and BA as radius, and any circle cutting this is described with D as centre. Shew that if a chord to both the circles be drawn
from A, through one of the points of intersection, it will be bisected by this point.
114. ABC is an acute-angled triangle, E and F are the middle points of the sides AB and AC. Shew that a line drawn from E, equal to EA, to meet the base, and another from F, equal to FA, also to meet it, will intersect the base at the same point.
Hence explain how, by folding a piece of paper such as the triangle ABC, it may be shewn that the three angles of a triangle are equal to two right angles.
115. A line AB is divided into any two parts in C, and on the whole line, and on the two parts of it, similar isosceles triangles, ADB, ACE, BCF, are described, the two latter being on the same side of the line, and the former on the opposite side; if G, H, K, be the centres of the circles inscribed in these triangles, prove that the angles AGH, BGK are equal respectively to ADC, BDC, and that GH is equal to GK.
116. Within a circle, whose diameter is AB, another circle is inscribed, touching the outer circle in A, and passing through its centre 0. From a point N, in AB, a line NQP is drawn, meeting the inner circle in Q, and the outer circle in P, AN being equal to one-sixth of AB. Prove that the duplicate ratio of NQ to NP is equal to the ratio of 2 to 5.
117. Describe a square, which shall be equal to the sum of a given square and a given rectangle, a side of the given square being greater than half the difference of the two sides containing the rectangle.
IN Book 1. Def. 7., it is laid down that a Plane Surface is one in which, if any two points be taken, the straight line between them lies wholly in that surface.
This definition should be extended by the addition of the following words, and if the straight line be produced, every point in the part produced will lie in the plane.
Euclid professes to prove this in the first Proposition of Book XI., which is thus enunciated: " one part of a straight line cannot be in a plane, and another part out of the plane."
But this has been assumed again and again in the proofs of earlier propositions; thus, for example, we have called a circle a plane figure, and having drawn any radius to a circle we have assumed that the radius, produced within the circumference, will meet the circumference.
From the extended definition of a Plane Surface it follows that a straight line, which meets a plane, must either lie entirely in that plane, or meet it in one point only; for if it met the plane in two points, it would lie entirely in the plane.
The Definitions given at the commencement of Book XI. relate partly to Plane Surfaces and partly to Solid Figures. By a slight change in the order in which they stand in the Greek text, we obtain the advantage of arranging them in accordance with this twofold division.
Relating to Plane Surfaces.
I. A Plane Surface is one in which, if any two points be taken, the straight line between them lies wholly in that surface; and if the straight line be produced, every point in the part produced will lie in the plane.
II. When a straight line is at right angles to every straight line in a plane which meets it, it is said to be perpendicular to the plane.
Note.- It will be shown in Prop. IV. that when a straight line is at right angles to each of two other straight lines in a plane, which meet it, it is at right angles to every other straight line in the plane which meets it.
III. A plane is perpendicular to a plane, when the straight lines, drawn in one of the planes perpendicular to the common section of the two planes, are perpendicular to the other plane.
IV. The inclination of a straight line to a plane is the acute angle, contained by that straight line and another, drawn from the point at which the first line meets the plane, to the point at which a perpendicular to the plane, drawn from any point of the first line above the plane, meets the same plane.
V. The inclination of a plane to a plane is the acute angle, contained by two straight lines, drawn from any the same point of their common section, at right angles to it, one in one plane, and the other in the other plane.
VI. Two planes are said to have the same inclination to one another, which two other planes have, when the said angles of inclination are equal to one another.
VII. Parallel planes are such as do not meet one another though produced.
Relating to Solid Figures.
VIII. A Solid is that which has length, breadth, and thick
IX. That which bounds a solid is a superficies.
X. A Solid Angle is that, which is made by the meeting of more than two plane angles, which are not in the same plane, at one point.
Definitions I. to X. are all that are required in the part of Book XI. included in this work. Those which follow are necessary to the explanation of some of the terms, which will be found in the Exercises and Examination Papers.
XI. Similar solid figures are such, as have all their solid angles equal, each to each, and are contained by the same number of similar planes.
XII. A Pyramid is a solid figure, contained by planes, which are constructed between one plane and one point above it, at which they meet.
XIII. A Prism is a solid figure, contained by plane figures, of which two that are opposite are equal, similar, and parallel to one another; and the others are parallelograms.
XIV. A Sphere is a solid figure, described by the revolution of a semicircle about its diameter, which remains fixed.
XV. The Axis of a Sphere is the fixed straight line, about which the semicircle revolves.
XVI. The Centre of a Sphere is the same with that of the semicircle.
XVII. The Diameter of a Sphere is any straight line, which passes through the centre, and is terminated both ways by the superficies of the sphere.
XVIII. A Cone is a solid figure, described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed. If the fixed side be equal to the other side containing the right angle, the cone is called a right-angled cone; if it be less than the other side, an obtuse-angled cone; and if greater, an acute-angled cone.
XIX. The Axis of a Cone is the fixed straight line, about which the triangle revolves.