An Elementary Treatise on Plane and Solid GeometryJ. Munroe, 1837 - 159 σελίδες |
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Σελίδα xvi
... fall upon SE without the triangle ADS , nor can it be within this tri- angle , as at H or at E , for AF + FS > AD + DS , whereas , by art . 40 , AH + HS < AD + DS . The angle ASB must , therefore , be greater than ASD , and ASB + BSC ...
... fall upon SE without the triangle ADS , nor can it be within this tri- angle , as at H or at E , for AF + FS > AD + DS , whereas , by art . 40 , AH + HS < AD + DS . The angle ASB must , therefore , be greater than ASD , and ASB + BSC ...
Σελίδα 9
... fall upon AE , and the angles will coincide . 30. Theorem . If two parallel lines AB , CD ( fig . 13 ) are cut by a third straight line EF , the external - internal angles , as EMB and END , or BMF and DNF , are equal , and the ...
... fall upon AE , and the angles will coincide . 30. Theorem . If two parallel lines AB , CD ( fig . 13 ) are cut by a third straight line EF , the external - internal angles , as EMB and END , or BMF and DNF , are equal , and the ...
Σελίδα 12
... fall upon DA , because the right angles CDB and CDA are equal ; the point F will fall upon E , because DF and DE are equal ; and the straight lines CF and CE will coincide . 39. Theorem . A perpendicular measures the short- est distance ...
... fall upon DA , because the right angles CDB and CDA are equal ; the point F will fall upon E , because DF and DE are equal ; and the straight lines CF and CE will coincide . 39. Theorem . A perpendicular measures the short- est distance ...
Σελίδα 16
... falling at once in each of the lines AC and BC , must fall upon their point of intersection C. The triangles will therefore coincide , and must be equal . 54. Corollary . Hence , when a side and the two adjacent angles of one triangle ...
... falling at once in each of the lines AC and BC , must fall upon their point of intersection C. The triangles will therefore coincide , and must be equal . 54. Corollary . Hence , when a side and the two adjacent angles of one triangle ...
Σελίδα 18
... fall within the other triangle ACB , as in fig . 19 , because , by art . 40 , AD + DB must in this case be less than ACCB . Secondly . If D falls without ACB , as in fig . 33 , the tri- angles ACD and BCD are isosceles , since AC is ...
... fall within the other triangle ACB , as in fig . 19 , because , by art . 40 , AD + DB must in this case be less than ACCB . Secondly . If D falls without ACB , as in fig . 33 , the tri- angles ACD and BCD are isosceles , since AC is ...
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Συχνά εμφανιζόμενοι όροι και φράσεις
ABCD &c AC AC adjacent angles altitude angle ABC angle BAC arc BC base and altitude bisect CD fig centre chord circumference convex surface Corollary cylinder DEF fig Definitions Demonstration denote diameter divided Draw equal arcs equal distances equilateral equivalent four right angles frustum given angle given circle given line given polygon given sides given square greater half the product Hence homologous sides hypothenuse infinitely small inscribed circle isoperimetrical isosceles Join AC Let ABCD line AB fig lines drawn mean proportional number of sides oblique lines parallel lines parallelogram parallelopiped perimeter perpendicular plane angles plane MN polygon ABCD prism Problem radii radius ratio rectangles regular polygon respectively equal right triangles Scholium secant sector segment side AC similar polygons similar triangles solid angle Solution sphere spherical polygon spherical triangle tangent Theorem triangles ABC triangular prism vertex vertices whence
Δημοφιλή αποσπάσματα
Σελίδα 148 - 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'.
Σελίδα 90 - To construct a parallelogram equivalent to a given square, and having the difference of its base and altitude equal to a given line.
Σελίδα 24 - CIRCLE is a plane figure bounded by a curved line, all the points of which are equally distant from a point within called the centre; as the figure ADB E.
Σελίδα 40 - One side and two angles of a triangle being given, to construct the triangle. Solution.
Σελίδα 79 - Construct, by § 145, a right triangle, of which the hypothenuse BC (fig. 79) is equal to the side of the greater square, and the leg AB is equal to the side of the less square ; and AC is the side of the required square.
Σελίδα 142 - THEOREM. If two triangles on the same sphere, or on equal spheres, are mutually equiangular, they will also be mutually equilateral. Let A and B be the two given triangles; P and Q their polar triangles. Since the angles are equal in the triangles A and B, the sides will be equal in. their polar triangles P and Q (Prop.
Σελίδα 6 - The preface and commentary to the Antigone are even more creditable to Mr. Woolsey's ability than those to the Alcestis. The sketch of the poem, in the preface, is written with clearness and brevity. The difficulties in this play, that call for a commentator's explanation, are far more numerous than in the Alcestis.
Σελίδα 70 - 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.
Σελίδα 5 - A surface is that which has length and breadth, without thickness. 6. A plane is a surface, in which any two points being taken, the straight line joining those points lies wholly in that surface.
Σελίδα 137 - Each side of a spherical triangle is less than the sum of 'the other two sides. 48. The sum of the sides of a spherical polygon is less than 360°.