Primary Elements of Plane And Solid Geometry for Schools And Academies
Primary Elements of Plane And Solid Geometry for Schools And Academies
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THEOREM IV.
Two triangles having the sides of the one successively proportional to the sides of the other, are similar.
BOOK II.
Gl Let ABC, DEF, be two triangles having AB : DE : : BC : EF : : CA : FD. It may be shown that they are similar.
On DE describe the triangle DGE, having the angles EDG, DEG, respect- ively equal to the angles A and B. Then will the triangles ABC and DEG be similar (Cor. , Theo. Ill) ; and we shall have (Def. 1, Sec. X) AB : DE : : BC : EG.
But by hypothesis we have AB... : DE : : BC : EF.
Hence, EG is equal to EF. In the same manner it may be shown that DG is equal to DF. Therefore, the two triangles DGE, DFE, are mutually equilateral and consequently equal (Cor. , Theo. XII, B. I). But ABC is by construction similar to DGE : hence, it is also similar to DFE.
Therefore, two triangles, etc.
THEOEEM V.
Two triangles having an angle of the one equal to an angle of the other, and the sides about those angles pro- portional, are similar.
Let the two triangles ABC, DEF, have the angle A equal to the angle D, and the sides AB, AC, proportional to 62 GEOMETRY.
What to read after Primary Elements of Plane And Solid Geometry for Schools And Academies?
You can find similar books in the "Read Also" column, or choose other free books by E W Evan Wilhelm Evans to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
THEOREM IV.
Two triangles having the sides of the one successively proportional to the sides of the other, are similar.
BOOK II.
Gl Let ABC, DEF, be two triangles having AB : DE : : BC : EF : : CA : FD. It may be shown that they are similar.
On DE describe the triangle DGE, having the angles EDG, DEG, respect- ively equal to the angles A and B. Then will the triangles ABC and DEG be similar (Cor. , Theo. Ill) ; and we shall have (Def. 1, Sec. X) AB : DE : : BC : EG.
But by hypothesis we have AB... : DE : : BC : EF.
Hence, EG is equal to EF. In the same manner it may be shown that DG is equal to DF. Therefore, the two triangles DGE, DFE, are mutually equilateral and consequently equal (Cor. , Theo. XII, B. I). But ABC is by construction similar to DGE : hence, it is also similar to DFE.
Therefore, two triangles, etc.
THEOEEM V.
Two triangles having an angle of the one equal to an angle of the other, and the sides about those angles pro- portional, are similar.
Let the two triangles ABC, DEF, have the angle A equal to the angle D, and the sides AB, AC, proportional to 62 GEOMETRY.
What to read after Primary Elements of Plane And Solid Geometry for Schools And Academies?
You can find similar books in the "Read Also" column, or choose other free books by E W Evan Wilhelm Evans to read onlineMoreLess
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