Elements of Geometry and Trigonometry from the Works of A.M. Legendre: Adapted to the Course of Mathematical Instruction in the United States |
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Page 49
... THEOREM . The diagonals of a parallelogram ( divide each other into equal parts , or mutually bisect each other ... Show that the lines which bisect ( BOOK I. 49.
... THEOREM . The diagonals of a parallelogram ( divide each other into equal parts , or mutually bisect each other ... Show that the lines which bisect ( BOOK I. 49.
Page 281
... show that the perimeter of PBF is greater than that of ABF . XIII . THEOREM . - Let an altitude of the triangle ABC be drawn from the vertex A , and also the bisectrix of the angle A ; then show that their included angle is equal to ...
... show that the perimeter of PBF is greater than that of ABF . XIII . THEOREM . - Let an altitude of the triangle ABC be drawn from the vertex A , and also the bisectrix of the angle A ; then show that their included angle is equal to ...
Page 284
... THEOREM . - Show that the bisectrices of the four angles of any quadrilateral intersect in four points , all of which lie on the circumference of the same circle . XLII . THEOREM . - If two circles touch each other exter- nally , and if ...
... THEOREM . - Show that the bisectrices of the four angles of any quadrilateral intersect in four points , all of which lie on the circumference of the same circle . XLII . THEOREM . - If two circles touch each other exter- nally , and if ...
Page 286
... THEOREM . - Show that the line which joins the middle points of two opposite sides of any quadrilateral , bisects the line joining the middle points of the two diagonals . LIV . THEOREM . - If from the extremities of one of the oblique ...
... THEOREM . - Show that the line which joins the middle points of two opposite sides of any quadrilateral , bisects the line joining the middle points of the two diagonals . LIV . THEOREM . - If from the extremities of one of the oblique ...
Page 287
... show that the difference of the squares of these segments is equal to the square of the other side about the right angle . LXI . THEOREM . - If lines are drawn from any point P to the four vertices of a rectangle , show that the sum of ...
... show that the difference of the squares of these segments is equal to the square of the other side about the right angle . LXI . THEOREM . - If lines are drawn from any point P to the four vertices of a rectangle , show that the sum of ...
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Common terms and phrases
ABCD AC² adjacent angles altitude apothem Applying logarithms bisects centre chord circle circumference circumscribed cone consequently convex surface cosec cosine Cotang cylinder denote diagonals diameter difference distance divided draw drawn edges equilateral feet find the area formula frustum given angle given line given point greater hence homologous hypothenuse included angle inscribed intersection isosceles less Let ABC log sin lower base lune mantissa number of sides parallel parallelogram parallelopipedon perimeter perpendicular plane angles plane MN polyedral angle polyedron principle demonstrated prism PROBLEM.-In PROPOSITION proved pyramid quadrant radii radius rectangle regular polygon right angles right-angled triangle Scholium secant segment semi-circumference similar sine slant height solution sphere spherical angle spherical polygon spherical triangle square straight line subtracting supplement Tang tangent THEOREM THEOREM.-Show tri-rectangular triangle ABC triangular prism triedral angle upper base vertex vertices volume whence
Popular passages
Page 28 - If two triangles have two sides of the one equal to two sides of the...
Page 90 - A cos 6 = cos a cos c + sin a sin c cos B cos c = cos a cos 6 + sin a sin 6 cos C Law of Cosines for Angles cos A = — cos B...
Page 18 - Things which are equal to the same thing, are equal to each other. 2. If equals are added to equals, the sums are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. If equals are added to unequals, the sums are unequal.
Page 4 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Page 253 - For, the point A being the pole of the arc EF, the distance AE is a 'quadrant ; the point C being the pole of the arc DE, the distance CE is likewise a quadrant : hence the point E is...
Page 61 - A Circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the center.
Page 94 - A'B'C', and applying the law of cosines, we have cos a' = cos b' cos c' + sin b' sin c' cos A'. Remembering the relations a' = 180° -A, b' = 180° - B, etc. (this expression becomes cos A = — cos B cos C + sin B sin C cos a.
Page 126 - If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar.
Page 46 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 125 - THEOREM. Similar triangles are to each other as the squares of their homologous sides.