A Second Book in Geometry

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Brewer and Tileston, 1863 - Geometry - 136 pages
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Page 47 - ... can be done, since a polygon of given sides is manifestly largest when most nearly circular, and a polygon of given perimeter inscribed in a circle is manifestly largest when the sides are equal. We can surely have no difficulty in proving these two points, and then our proof will be complete. 1 33. Let us return, then, to the synthetic mode, and establish these propositions : — First, that the maximum of polygons formed of given sides may be inscribed in a circle ; secondly, that the maximum...
Page 36 - С,) which is impossible, and therefore the angles cannot be altered. 95. Corollary. If the three sides of a triangle are respectively equal to the three sides of another triangle, the angles of one must be equal to those of the other, and the equal angles are enclosed in the equal sides. 96. Theorem. If the opposite sides of a quadrangle are equal, the quadrangle is a parallelogram. — Proof. If in the quadrangle А В С D (Fig.
Page 36 - AB are equal to the adjacent angles CAD and ACD; whence, by article 91, the two triangles are equal, and AD is equal to BC, and AB equal to D C. 93. Axiom. If one end of a straight line stands still while the other turns round, the end that moves will begin to move in a direction at right angles to that of the lino itself.
Page 24 - ... these are self-evident truths. 51. By a self-evident step in reasoning, I mean the statement of the relation of one truth to another, or of the dependence of one truth upon another, when that dependence or that relation is itself a self-evident truth.
Page 31 - We may afterwards, if we like, devise other modes of analysis and synthesis ; for this proposition, like all others, may be approached in various ways. 65. The Pythagorean proposition or theorem might be suggested in different ways. But in whatever way we were led to suspect that the square on the hypothenuse is equivalent to the sum of the squares on the legs, we should, in reflecting upon it, probably begin by drawing a right triangle with a square built upon each side. 66. We should inquire whether...
Page 34 - N, and this will give us MXQ : NXQ = NXP : NX Q. Thus from the self-evident truth of article 78 we find that the product of the means bears the same ratio to the product NXQ that is borne to it by the product of the extremes. And as it is self-evident that two quantities, bearing the same ratio to a third, must be equal to each other, we have proved that the product of the means is equal to that of the extremes. 82. Definition. When both the means are the same quantity, that quantity is called a...
Page 39 - ... to make the remainder small enough to be neglected. 101. Definitions. The right angle, right triangle, legs, and hypotenuse, are defined in articles 14 and 17. 102. Theorem. The sum of the three angles of a triangle is equivalent to two right angles. This proposition has been proved in articles 26-31, 34-36, and 57-62. 103. Corollary. The sum of the two angles opposite to the legs of a right triangle is equivalent to one right angle. 104. Corollary. If an angle opposite a leg in one right triangle...
Page 41 - Theorem. The square on the hypothenuse is equivalent to the sum of the squares on the legs. Proof. Having drawn the figure (Fig. A.), as for the former proof, draw the lines C' B, and B
Page 34 - Proof. For as the straight line has but one direction, and each of the parallel lines may always be considered as going in the same direction as the other, the difference of that direction from the direction of the third straight line must be the same for each of the parallel lines. 88. Corollary. If a straight line is parallel to one of two parallel lines, it is parallel to the other ; if at right angles to one of the two, it is at right angles to the other. 89. Theorem. If a straight line make...
Page 132 - If two triangles on the same sphere, or on equal spheres, are mutually equiangular, they will also be mutually equilateral. Let A and B be the two given triangles; P and Q their polar triangles. Since the angles are equal in the triangles A and B, the sides will be equal in.

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