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altitude applied base called centre chord circle circumference coincide common construct contained convex surface Corollary cylinder Definitions denote described diameter difference divided Draw equal equal distances equiangular equilateral equivalent extremities faces fall figure formed four frustum given given polygon given square greater half the product Hence homologous sides included infinitely small inscribed intersection isosceles Join less maximum mean measure meet middle number of sides opposite parallel parallel planes parallelogram parallelopipeds passes perimeter perpendicular plane pole polyedron preceding prism Problem Proof proportional prove pyramid radii radius ratio rectangles regular polygon respectively right angles right triangle sector segment side BC similar similar polygons solid angle solidity Solution sphere spherical triangle square straight line Suppose surface Take tangent Theorem third triangles ABC vertex vertices whence
Page 133 - The sum of the angles of a spherical triangle is greater than two and less than six right angles ; that is, greater than 180° and less than 540°. (gr). If A'B'C' is the polar triangle of ABC...
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 126 - A sphere is a solid terminated by a curved surface, all the points of which are equally distant from a point within called the centre.
Page 87 - To construct a parallelogram equivalent to a given square, and having the difference of its base and altitude equal to a given line.
Page 16 - Theorem. In an isosceles triangle the angles opposite the equal sides are equal.
Page 141 - The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. D c A' D' Hyp. In triangles ABC and A'B'C', ZA = ZA'. To prove AABC = ABxAC. A A'B'C' A'B'xA'C' Proof. Draw the altitudes BD and B'D'.
Page 136 - ADC ; the last two are therefore right angles ; hence the arc drawn from the vertex of an isosceles spherical triangle to the middle of the base, is perpendicular to the base, and bisects the vertical angle.
Page 122 - The volume of the frustum of any pyramid is equal to the sum of the volumes of three pyramids whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and a mean proportional between the bases of the frustum.