Elements of plane geometry, book i, containing nearly the same propositions as the first book of Euclid's Elements1865 |
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Seite 37
... reason , CD cannot fall on GL . Since therefore CD can neither fall on the one side of GH nor the other , CD must fall upon it ; and therefore the right angles at the points C and G are all equal , viz . , the angle ACD to EGH , and BCD ...
... reason , CD cannot fall on GL . Since therefore CD can neither fall on the one side of GH nor the other , CD must fall upon it ; and therefore the right angles at the points C and G are all equal , viz . , the angle ACD to EGH , and BCD ...
Seite 40
... reason , AB and BC are greater than AC , and AC and CB greater than AB ; and therefore any two sides of a triangle are together greater than the third side . B C PROPOSITION IX . LEMMA . The circumferences of circles cannot cut one ...
... reason , AB and BC are greater than AC , and AC and CB greater than AB ; and therefore any two sides of a triangle are together greater than the third side . B C PROPOSITION IX . LEMMA . The circumferences of circles cannot cut one ...
Seite 42
... reason , FE is equal to C , and , by construction , DE is equal to B ; therefore the triangle FDE has its three sides equal to the three given straight lines A , B , and C , each to each . It is also evident , that if the lines A and C ...
... reason , FE is equal to C , and , by construction , DE is equal to B ; therefore the triangle FDE has its three sides equal to the three given straight lines A , B , and C , each to each . It is also evident , that if the lines A and C ...
Seite 49
... reason , the exterior angle CEB of the triangle ABE is greater than BAC ; and it has been demonstrated that the angle BDC is greater than the angle CEB , much more then is the angle BDC greater than the angle BAC . PROPOSITION XXII ...
... reason , the exterior angle CEB of the triangle ABE is greater than BAC ; and it has been demonstrated that the angle BDC is greater than the angle CEB , much more then is the angle BDC greater than the angle BAC . PROPOSITION XXII ...
Seite 60
... reason , the angle GAB is equal G Ak G C E D F F to GDE ; and by adding equals to equals , or taking equals from equals , the whole or the part BAC is equal to the whole or the part DEF . PROPOSITION XXXVIII . THEOR . Straight lines ...
... reason , the angle GAB is equal G Ak G C E D F F to GDE ; and by adding equals to equals , or taking equals from equals , the whole or the part BAC is equal to the whole or the part DEF . PROPOSITION XXXVIII . THEOR . Straight lines ...
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Elements of Plane Geometry, Book: Containing Nearly the Same Propositions As ... Euclid Keine Leseprobe verfügbar - 2008 |
Häufige Begriffe und Wortgruppen
AB is equal ABC and DEF ABC is equal acute adjacent angles ancient geometers angle ACD angle AGH angle BAC angles ABC angles equal angular magnitude base BC bisect centre circumference coincide diagonal drawn EBCF equal alternate angles equal Def equal to BC Euclid EUCLID'S ELEMENTS exterior angle figure has sides four right angles geometers given point given straight line greater than AC included angle interior opposite angle intersect isosceles triangle join less Let ABC Let the straight method method of exhaustions parallel lines parallel to CD parallelogram ABCD perpendicular PLANE GEOMETRY point F PROB proof properties of parallel PROPOSITION Pythagoras radius rectangle rectilineal figure reductio ad absurdum Scholium side AB side AC straight line BC THEOR theorem three angles three sides triangle ABC triangle DEF triangles are equal truths unequal vertex vertical angle wherefore
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Seite 43 - If two triangles have two sides and the included angle of the one, equal to two sides and the included angle of the other, each to each, the two triangles will be equal.
Seite 46 - Any two angles of a triangle are together less than two right angles.
Seite 37 - The angles which one straight line makes with another upon one tide of it, are either two right angles, or are together equal to two right angles. Let the straight line AB make with CD, upon one side of it the angles CBA, ABD ; these are either two right angles, or are together equal to two right angles. For, if the angle CBA be equal to ABD, each of them is a right angle (Def.
Seite 57 - Through a given point to draw a straight line parallel to a given straight line, Let A be the given point, and BC the given straight line : it is required to draw through the point A a straight line parallel to BC.
Seite 38 - ... in one and the same straight line. At the point B in the straight line AB, let the two straight lines BC, BD upon the opposite sides of AB, make the adjacent angles ABC, ABD, equal together to two right angles. BD is in the same straight line with CB.
Seite 68 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Seite 34 - LET it be granted that a straight line may be drawn from any one point to any other point.
Seite 64 - Parallelograms upon the same base, and between the same parallels, are equal to one another.
Seite 46 - If one side of a triangle be produced, the exterior angle is greater than either of the interior, and opposite angles.
Seite 34 - Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal.