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 26
... triangles coincide throughout , therefore equal in all their parts ( I. , D. 14 ) ; which was to be proved . 1 PROPOSITION VII . THEOREM . The sum of any two sides of a triangle is greater than the third side . Let ABC be a triangle ...
... triangles coincide throughout , therefore equal in all their parts ( I. , D. 14 ) ; which was to be proved . 1 PROPOSITION VII . THEOREM . The sum of any two sides of a triangle is greater than the third side . Let ABC be a triangle ...
Page 28
... triangle ABC it may be on the side BC , or it may be within the tri angle ABC . Each case will be considered separately . 1o . When G is without the triangle ABC . In the triangles GIC and AIB , we have , B ( P. VII . ) , G A D F E GI + ...
... triangle ABC it may be on the side BC , or it may be within the tri angle ABC . Each case will be considered separately . 1o . When G is without the triangle ABC . In the triangles GIC and AIB , we have , B ( P. VII . ) , G A D F E GI + ...
Page 29
... triangle ABC . From Proposition VIII . , we have , BA + BC > GA + GC ; = or , since GA BA , and GC EF , we have , = BA + BC > BA + EF . Taking away the common part AB , A D B G E there remains , BC > EF . F Hence , in each case , BC is ...
... triangle ABC . From Proposition VIII . , we have , BA + BC > GA + GC ; = or , since GA BA , and GC EF , we have , = BA + BC > BA + EF . Taking away the common part AB , A D B G E there remains , BC > EF . F Hence , in each case , BC is ...
Page 30
... triangles have the three sides of the one equal to the three sides of the other , each to each , the triangles will be equal in all their parts . In the triangles ABC and DEF , let AB be equal to DE , AC to DF , and BC to EF : angles be ...
... triangles have the three sides of the one equal to the three sides of the other , each to each , the triangles will be equal in all their parts . In the triangles ABC and DEF , let AB be equal to DE , AC to DF , and BC to EF : angles be ...
Page 31
... triangle are equal , the sides opposite to them are also equal , and consequently , the triangle is isos- celes . In the triangle ABC , let the angle ABC be equal to the angle ACB : then will AC be equal to AB , and consequently , the ...
... triangle are equal , the sides opposite to them are also equal , and consequently , the triangle is isos- celes . In the triangle ABC , let the angle ABC be equal to the angle ACB : then will AC be equal to AB , and consequently , the ...
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Common terms and phrases
AB² AC² altitude angle ACB apothem axis base and altitude base multiplied BC² bisect centre chord circumference coincide cone consequently convex surface corresponding cosec cosine Cotang cylinder denote diameter distance divided draw drawn edges equal bases equal in volume equal to AC equal to half equally distant Formula frustum given angle given line greater hence homologous hypothenuse included angle intersection less Let ABC logarithm lower base mantissa mean proportional measured by half number of sides opposite parallelogram perimeter perpendicular plane MN polyedral angle polyedron prism PROPOSITION XI proved pyramid quadrant radii radius rectangle regular polygons right-angled triangle Scholium segment semi-circumference side BC similar sine slant height sphere spherical angle spherical excess spherical polygon spherical triangle straight line tangent THEOREM triangle ABC triangular prisms triedral angle upper base vertex vertices whence