Elements of Geometry |
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Page 7
... equal to two sides and the included angle of the other , each to each . Demonstration . In the two triangles ABC , DEF ( fig . 2. ) , let Fig . 23- the angle A be equal to the angle D , the side AB equal to the side DE , and the side AC ...
... equal to two sides and the included angle of the other , each to each . Demonstration . In the two triangles ABC , DEF ( fig . 2. ) , let Fig . 23- the angle A be equal to the angle D , the side AB equal to the side DE , and the side AC ...
Page 9
... equal AG , and there remains GC < BC ; therefore EF < BC . Case 11. If the point G ( fig . 26 ) fall upon the side BC , then Fig . 26 . it is evident that GC , or its equal EF , is less than BC . Case III . If the point G ( fig . 27 ) ...
... equal AG , and there remains GC < BC ; therefore EF < BC . Case 11. If the point G ( fig . 26 ) fall upon the side BC , then Fig . 26 . it is evident that GC , or its equal EF , is less than BC . Case III . If the point G ( fig . 27 ) ...
Page 10
... equal to one of the others . Fig . 20 . Fig . 30 . THEOREM . 48. Reciprocally , if two angles of a triangle are equal , the oppo- site sides are equal , and the triangle is isosceles . Demonstration . Let the angle ABC = ACB ( fig . 29 ) ...
... equal to one of the others . Fig . 20 . Fig . 30 . THEOREM . 48. Reciprocally , if two angles of a triangle are equal , the oppo- site sides are equal , and the triangle is isosceles . Demonstration . Let the angle ABC = ACB ( fig . 29 ) ...
Page 11
... equal to the triangle ABC . For the angle CBF is a right angle ( 29 ) , as well as CBA , and the side BF = BA ; therefore the triangles are equal ( 36 ) , and the angle BCF BCA . But BCA is , by hypothesis , a right angle ; there- fore ...
... equal to the triangle ABC . For the angle CBF is a right angle ( 29 ) , as well as CBA , and the side BF = BA ; therefore the triangles are equal ( 36 ) , and the angle BCF BCA . But BCA is , by hypothesis , a right angle ; there- fore ...
Page 12
Adrien Marie Legendre. = 1. The triangle BCF is equal to the triangle BCA ; for the right angle CBF = CBA , the side CB is common , and the side BF BA ; therefore the third side CF is equal to the third side AC ( 36 ) . But AF < AC + CF ...
Adrien Marie Legendre. = 1. The triangle BCF is equal to the triangle BCA ; for the right angle CBF = CBA , the side CB is common , and the side BF BA ; therefore the third side CF is equal to the third side AC ( 36 ) . But AF < AC + CF ...
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Common terms and phrases
ABC fig adjacent angles altitude angle ACB angle BAD base ABCD bisect centre chord circ circular sector circumference circumscribed common cone consequently construction convex surface Corollary cube cylinder Demonstration diagonals diameter draw drawn equal and parallel equiangular equilateral equivalent faces figure four right angles frustum Geom gles greater hence homologous sides hypothenuse inclination inscribed circle isosceles join less let fall line AC manner mean proportional measure the half meet multiplied number of sides oblique lines opposite parallelogram parallelopiped perimeter perpendicular plane MN polyedron prism proposition pyramid S-ABC quadrilateral radii radius ratio rectangle regular polygon right angles right-angled triangle Scholium segment semicircumference side BC similar solid angle sphere spherical polygons spherical triangle square described straight line tangent THEOREM three plane angles triangle ABC triangular prism triangular pyramids vertex vertices whence