First principles of Euclid: an introduction to the study of the first book of Euclid's Elements1880 |
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Σελίδα 30
... DFE . ( Euc . I. I shows how to do this . ) Join A F. The line A F shall bisect the angle B A С. Π B If the angle BAC is bisected by AF ; the angle DAF will be equal to the angle E A F. ... We have to prove that angle DAF equals angle ...
... DFE . ( Euc . I. I shows how to do this . ) Join A F. The line A F shall bisect the angle B A С. Π B If the angle BAC is bisected by AF ; the angle DAF will be equal to the angle E A F. ... We have to prove that angle DAF equals angle ...
Σελίδα 71
... angle A B C is equal to the angle DE F ; ( d ) The angle ACB is equal to the angle DFE . B A E D F Proof ( by superposition ) . ( The different steps of this proof can be more readily made clear by pasteboard triangles , one of which ...
... angle A B C is equal to the angle DE F ; ( d ) The angle ACB is equal to the angle DFE . B A E D F Proof ( by superposition ) . ( The different steps of this proof can be more readily made clear by pasteboard triangles , one of which ...
Σελίδα 72
... angle DE F. ( f ) ... angle ABC is equal to angle DEF , also , angle A CD coincides with angle DFE . ( g ) ... angle ACD is equal to angle DFE . Collecting ( d ) , ( e ) , ( f ) , and ( g ) , we have The base BC equal to the base E F ...
... angle DE F. ( f ) ... angle ABC is equal to angle DEF , also , angle A CD coincides with angle DFE . ( g ) ... angle ACD is equal to angle DFE . Collecting ( d ) , ( e ) , ( f ) , and ( g ) , we have The base BC equal to the base E F ...
Σελίδα 85
... angle BAC is equal to angle EDF . Let us suppose that AB is not equal to DE ; then , one of the two must be greater ... DFE . Again , ( k ) angle ACB is greater than angle GCB , but angle ACB is equal to angle DFE ( Given b ) . And ...
... angle BAC is equal to angle EDF . Let us suppose that AB is not equal to DE ; then , one of the two must be greater ... DFE . Again , ( k ) angle ACB is greater than angle GCB , but angle ACB is equal to angle DFE ( Given b ) . And ...
Σελίδα 86
... angle A CB is equal to its part GCB : which is im- possible ( Axiom 9 ) . ... A B is not unequal to DE , that is , ( m ) AB is equal to DE , and BC is equal to EF , and the included angle ABC is equal to the included angle DEF ( Given a ) ...
... angle A CB is equal to its part GCB : which is im- possible ( Axiom 9 ) . ... A B is not unequal to DE , that is , ( m ) AB is equal to DE , and BC is equal to EF , and the included angle ABC is equal to the included angle DEF ( Given a ) ...
Συχνά εμφανιζόμενοι όροι και φράσεις
A B D A B equal A B is parallel ABC is equal ACD is greater adjacent angles alternate angle angle A B C angle ABC angle ACB angle AGH angle B A C angle BAC angle BCD angle contained angle DFE angle EDF angle GHD angles CBE angles equal Axiom 2a Axiom 9 BAC is equal base BC bisected Construction definition enunciations of Euc equal angles equal to angle equal to BC equilateral triangle EXERCISES.-I exterior angle figure given line given straight line greater than angle included angle interior opposite angle isosceles triangle Join Let us suppose line CD major premiss parallel to CD parallelogram Particular Enunciation PROBLEM Euclid produced proposition prove that 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 142 - 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 133 - 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.