The Elements of Geometry
Leach, Shewell & Sanborn, 1894 - Geometry - 378 pages
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ABCD adjacent altitude axis base bisector bisects called centre chord circle circumference circumscribed coincide common cone construct cylinder denote describe diagonals diameter diedral divided draw drawn equal respectively equally distant equilateral equivalent EXERCISES extremities faces fall figure Find formed four frustum given given point greater Hence homologous hypotenuse included inscribed intersection isosceles triangle joining lateral area length less line drawn manner mean measured meet middle point multiplied parallel parallelogram parallelopiped passing perimeter perpendicular plane polygon prism PROBLEM produced Prop proportional PROPOSITION prove pyramid quadrilateral radii radius ratio rectangle regular respectively right angles right triangle secant segment sides similar sphere spherical triangle square straight line surface tangent tetraedron THEOREM third trapezoid triangle ABC unit vertex vertices volume Whence
Page 38 - If two triangles have two sides of the one equal respectively to two sides of the other, but the included angle of the first greater than the included angle of the second, then the third side of the first is greater than the third side of the second. Given A ABC and A'B'C' with UNEQUAL LINES AND UNEQUAL ANGLES Proof STATEMENTS Apply A A'B'C' to A ABC so that A'B
Page 146 - In any triangle, the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides, minus twice the product of one of these sides and the projection of the other side upon it 190 THEOREM XLIX 196.
Page 140 - If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar.
Page 122 - In any proportion the terms are in proportion by Composition ; that is, the sum of the first two terms is to the first term as the sum of the last two terms is to the third term.
Page 65 - The straight line joining the middle points of two sides of a triangle is parallel to the third side, and equal to half of it.
Page 172 - 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. To prove that Proof. A Let the triangles ABC and ADE have the common angle A. A ABC -AB X AC Now and A ADE AD X AE Draw BE.
Page 355 - Similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the diameters of their bases.
Page 123 - In any proportion the terms are in proportion by composition and division ; that is, the sum of the first two terms is to their difference as the sum of the last two terms to their difference.
Page 342 - The area of a lune is to the surface of the sphere as the angle of the lune is to four right angles, or as the arc which measures that angle is to the circumference.
Page 135 - ... they have an angle of one equal to an angle of the other and the including sides are proportional; (c) their sides are respectively proportional.