An Elementary Treatise on Plane and Solid Geometry |
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Page 7
... In the same way , APC and BPC may be proved to be supplements of each other ; and therefore the vertical Adjacent and Vertical Angles . Sum of all the Angles 1 CH . IV . § 23. ] THE ANGLE . Complement and supplement of angle (22),
... In the same way , APC and BPC may be proved to be supplements of each other ; and therefore the vertical Adjacent and Vertical Angles . Sum of all the Angles 1 CH . IV . § 23. ] THE ANGLE . Complement and supplement of angle (22),
Page 10
... proved equal to MND . 31. Theorem . If two straight lines , lying in the same plane , as AB , CD ( fig . 13 ) , are cut by a third , EF , so that the angles EMB and END are equal , or AMN and MND are equal , & c .; the lines AB , CD ...
... proved equal to MND . 31. Theorem . If two straight lines , lying in the same plane , as AB , CD ( fig . 13 ) , are cut by a third , EF , so that the angles EMB and END are equal , or AMN and MND are equal , & c .; the lines AB , CD ...
Page 23
... proved that BAD = BCD . 79. Corollary . Two parallel lines comprehended between two other parallel lines are equal . " = 80. Theorem . If , in a quadrilateral ABCD ( fig . 39 ) , the opposite sides are equal , namely , AB CD , and AD BC ...
... proved that BAD = BCD . 79. Corollary . Two parallel lines comprehended between two other parallel lines are equal . " = 80. Theorem . If , in a quadrilateral ABCD ( fig . 39 ) , the opposite sides are equal , namely , AB CD , and AD BC ...
Page 36
... prove that the line ABM , perpendicular to the common tangent at M , passes through both the centres A and B. Position of a Point in a plane . CHAPTER IX 36 遷 PLANE GEOMETRY . [ CH . VIII . § 127 . Lines parallel to a third line (36),
... prove that the line ABM , perpendicular to the common tangent at M , passes through both the centres A and B. Position of a Point in a plane . CHAPTER IX 36 遷 PLANE GEOMETRY . [ CH . VIII . § 127 . Lines parallel to a third line (36),
Page 43
... proved that OF OD OE . = - AOE , by Hence the circumference DFE passes through the points D , F , E , and the sides are tangents to it , by § 120 . 153. Corollary . The three lines AO , BO , and CO , which bisect the three angles of a ...
... proved that OF OD OE . = - AOE , by Hence the circumference DFE passes through the points D , F , E , and the sides are tangents to it , by § 120 . 153. Corollary . The three lines AO , BO , and CO , which bisect the three angles of a ...
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Common terms and phrases
ABC fig adjacent angles angle BAC arc BC base and altitude bisect centre chord circumference common altitude construct convex surface Corollary DEF fig Definitions denote diameter divided Draw equal arcs equal distances equiangular with respect equilateral equivalent frustum given angle given circle given line given polygon given sides given square gles greater half the product Hence homologous sides hypothenuse infinite number infinitely small Inscribed Angle inscribed circle isosceles Let ABCD line AB fig line BC lines drawn mean proportional number of sides oblique lines parallel lines parallel to BC parallelogram parallelopipeds perimeter perpendicular plane MN polyedron polygon ABCD &c Problem Proof pyramid or cone radii radius rectangles regular polygon right triangle Scholium sector segment side BC similar polygons similar triangles solid angle Solution sphere spherical polygon spherical triangle straight line tangent Theorem triangles ABC triangular prism vertex vertices whence
Popular passages
Page 68 - The perimeters of two regular polygons of the same number of sides, are to each other as their homologous sides, and their areas are to each other as the squares of those sides (Prop.
Page 127 - Every section of a sphere, made by a plane, is a circle, Let AMB be a section, made by a plane, in the sphere whose centre is C.
Page 71 - Rectangles of the same altitude are to each other as their bases, and rectangles of the same base are to each other as their altitudes. 245.
Page 20 - The sum of the three angles of any triangle is equal to two right angles.
Page xv - The first term of a ratio is called the antecedent, and the second term the consequent.
Page 83 - ... we suppose the error A to be of any magnitude whatever. 286. Definition. Similar sectors and similar segments are such as correspond to similar arcs. 287. Theorem. Similar sectors are to each other as the squares of their radii. Proof. The similar sectors AOB, A'OB ' (fig. 136) are, by the same reasoning as in t5 97, the same parts of their respective circles, which the angle O= O...
Page 31 - Theorem. In the same circle, or in equal circles, equal arcs are subtended by equal chords.
Page 87 - To construct a parallelogram equivalent to a given square, and having the sum of its base and altitude equal to a given line.
Page 99 - B, from the plane. 320. Theorem. Oblique lines drawn from a point to a plane at equal distances from the perpendicular are equal; and of two oblique lines unequally distant the more remote is the greater.
Page 78 - Similar triangles are to each other as the squares of their homologous sides. Proof. In the similar .triangles ABC, A'B'C