First principles of Euclid: an introduction to the study of the first book of Euclid's Elements1880 |
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Σελίδα 59
... angles , each to each , to which the equal sides are opposite ( Euc . I. 4 ) . .. the angle B A E is equal to the angle E C F. But the angle ECD is greater than the angle ECF ( Axiom 9 ) . .. The angle ACD is greater than the angle BAC ...
... angles , each to each , to which the equal sides are opposite ( Euc . I. 4 ) . .. the angle B A E is equal to the angle E C F. But the angle ECD is greater than the angle ECF ( Axiom 9 ) . .. The angle ACD is greater than the angle BAC ...
Σελίδα 60
... than two right angles . Construction . We will prove that angles A B C and A CB , are together less than two right angles . Produce the side B C to D. B Proof ( by Euc . I. 16 ) . D The exterior angle of any triangle is greater than ...
... than two right angles . Construction . We will prove that angles A B C and A CB , are together less than two right angles . Produce the side B C to D. B Proof ( by Euc . I. 16 ) . D The exterior angle of any triangle is greater than ...
Σελίδα 61
... greater side of every triangle has the greater angle opposite to it . A B Particular Enunciation . Given . The triangle A B C , having the side A C greater than AB . Required . To prove that angle ABC is greater than Theorem ( Euclid I. 18 ) ...
... greater side of every triangle has the greater angle opposite to it . A B Particular Enunciation . Given . The triangle A B C , having the side A C greater than AB . Required . To prove that angle ABC is greater than Theorem ( Euclid I. 18 ) ...
Σελίδα 62
... ) . Q E. D. EXERCISES . - I . In the figure of this proposition let the side B C be greater than the side A B. Then prove that angle B A C is greater than angle ACB . II . In the triangle ABC the angle BAC is 62 First Principles of Euclid .
... ) . Q E. D. EXERCISES . - I . In the figure of this proposition let the side B C be greater than the side A B. Then prove that angle B A C is greater than angle ACB . II . In the triangle ABC the angle BAC is 62 First Principles of Euclid .
Σελίδα 63
T S. Taylor. II . In the triangle ABC the angle BAC is bisected by the line AD . Prove that BA is greater than BD ; and CA greater than CD . ( Notice that B ADC is the exterior angle of the triangle A B D , and therefore ( Euc . I. 16 ) ...
T S. Taylor. II . In the triangle ABC the angle BAC is bisected by the line AD . Prove that BA is greater than BD ; and CA greater than CD . ( Notice that B ADC is the exterior angle of the triangle A B D , and therefore ( Euc . I. 16 ) ...
Συχνά εμφανιζόμενοι όροι και φράσεις
1st conclusion 2nd Syllogism A B equal ABC is equal adjacent angles alternate angle angle A CD angle ABC angle B A C angle BAC angle contained angle DFE angle EDF angle GHD angles BGH angles equal Axiom 2a Axiom 9 base B C bisected CD is greater coincide Construction definition diameter enunciations of Euc equal angles equal to A B equal to angle equal to CD equal to side equilateral triangle EXERCISES.-I exterior angle figure given line given point given straight line greater than angle included angle interior opposite angle isosceles triangle Join Let us suppose line A B line CD major premiss parallel to CD parallelogram Particular Enunciation PROBLEM Euclid produced proposition prove that angle remaining angle Required right angles side A C sides equal square THEOREM Euclid triangle ABC
Δημοφιλή αποσπάσματα
Σελίδα 83 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...
Σελίδα 18 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
Σελίδα 66 - If, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than, the other two sides of the triangle, but shall contain a greater angle. Let...
Σελίδα 34 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a Right Angle; and the straight line which stands on the other is called a Perpendicular to it.
Σελίδα 94 - Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of' the base, equal to one another, and likewise those which are terminated in the other extremity.
Σελίδα 88 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.
Σελίδα 104 - If a straight line falling upon two other straight lines, make the exterior angle equal to the interior and opposite upon the same side of the line ; or make the interior angles upon the same side together equal to two right angles ; the two straight lines shall be parallel to one another.
Σελίδα 140 - If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it ; the angle contained by these two sides is a right angle.
Σελίδα 51 - If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line.
Σελίδα 132 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.