An Elementary Treatise on Plane and Solid Geometry |
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Page vi
... included by two other lines ( 40 ) , Distances of a point from the extremities of a line ( 42 ) , CHAPTER VII . SIDES AND ANGLES OF POLYGONS , • 9 9 9 9 10 10 10 11 · 11 11 · 12 · 12 13 Plane figure , polygon , and its perimeter ( 43 ) ...
... included by two other lines ( 40 ) , Distances of a point from the extremities of a line ( 42 ) , CHAPTER VII . SIDES AND ANGLES OF POLYGONS , • 9 9 9 9 10 10 10 11 · 11 11 · 12 · 12 13 Plane figure , polygon , and its perimeter ( 43 ) ...
Page 12
... , similarly drawn , but included by them . Proof . Produce DA to E. We have , by § 18 , and AC + CE > AD + DE , DEBE > DB . Oblique Lines unequally Distant from the Perpendicular . The sum 12 [ CH . VI . § 40 . PLANE GEOMETRY .
... , similarly drawn , but included by them . Proof . Produce DA to E. We have , by § 18 , and AC + CE > AD + DE , DEBE > DB . Oblique Lines unequally Distant from the Perpendicular . The sum 12 [ CH . VI . § 40 . PLANE GEOMETRY .
Page 15
... included angle of the one are respective- ly equal to two sides and the included angle of the other . Proof . In the two triangles ABC , DEF , ( fig . 31 ) , let the angle A be equal to the angle D , and the sides AB , AC , respectively ...
... included angle of the one are respective- ly equal to two sides and the included angle of the other . Proof . In the two triangles ABC , DEF , ( fig . 31 ) , let the angle A be equal to the angle D , and the sides AB , AC , respectively ...
Page 16
... included angle of one triangle are respectively equal to those of another , the other side and angles are also equal in the two triangles . 53. Theorem . Two triangles are equal , when a side and the two adjacent angles of one triangle ...
... included angle of one triangle are respectively equal to those of another , the other side and angles are also equal in the two triangles . 53. Theorem . Two triangles are equal , when a side and the two adjacent angles of one triangle ...
Page 17
... included angle ABD , respectively equal to the two sides BC , BD , and the included angle DBC ; and the angle A will be equal to C. 56. Corollary . An equilateral triangie is also equi- angular . 57. Theorem . The line BD ( fig . 32 ) ...
... included angle ABD , respectively equal to the two sides BC , BD , and the included angle DBC ; and the angle A will be equal to C. 56. Corollary . An equilateral triangie is also equi- angular . 57. Theorem . The line BD ( fig . 32 ) ...
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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