Elements of plane geometry, book i, containing nearly the same propositions as the first book of Euclid's Elements |
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Seite 30
Hence two straight lines can neither enclose a space nor have a common segment ; for if a second straight line were drawn between the same points , it must coincide in every part and become one with the first . Cor . 2.
Hence two straight lines can neither enclose a space nor have a common segment ; for if a second straight line were drawn between the same points , it must coincide in every part and become one with the first . Cor . 2.
Seite 32
The radius of a circle is a straight line drawn from the centre to the circumference . Cor . Hence all the radii of the same circle are equal , for each of them is equal to the radiant . a a 35. The DIAMETER of a circle is a 32 ELEMENTS ...
The radius of a circle is a straight line drawn from the centre to the circumference . Cor . Hence all the radii of the same circle are equal , for each of them is equal to the radiant . a a 35. The DIAMETER of a circle is a 32 ELEMENTS ...
Seite 39
Hence also all the simple angles made by any number of straight lines meeting in one point are together equal to four right angles . PROPOSITION VIII . THEOR . Any two sides of a triangle are together greater than the third side .
Hence also all the simple angles made by any number of straight lines meeting in one point are together equal to four right angles . PROPOSITION VIII . THEOR . Any two sides of a triangle are together greater than the third side .
Seite 45
Hence every equilateral triangle is also equiangular . Cor . 2. The straight line bisecting the vertical angle of an isosceles triangle bisects also the base at right angles ; for the adjacent angles ADB and ADC are equal , and BD is ...
Hence every equilateral triangle is also equiangular . Cor . 2. The straight line bisecting the vertical angle of an isosceles triangle bisects also the base at right angles ; for the adjacent angles ADB and ADC are equal , and BD is ...
Seite 47
... because the side AB is equal to AD ; therefore the angle ABD is also greater than the angle ACB ; wherefore much more is the angle ABC greater than ACB . Cor . Hence the less angle of every triangle is opposite to the less side .
... because the side AB is equal to AD ; therefore the angle ABD is also greater than the angle ACB ; wherefore much more is the angle ABC greater than ACB . Cor . Hence the less angle of every triangle is opposite to the less side .
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Elements of Plane Geometry, Book: Containing Nearly the Same Propositions As ... Euclid Keine Leseprobe verfügbar - 2008 |
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ABC is equal AC is equal acute adjacent alternate angles ancient angle ACD angle BAC angles ABC appears application assume axiom base BC bisect called centre circle circumference coincide common conclusion construction definition demonstration describe determined diagonal draw drawn elements employed established Euclid extended exterior angle extremities fall four right angles geometers geometry given straight line greater half Hence included angle interior opposite angle intersect introduced join knowledge less Let ABC magnitudes manner means meet method mind mode necessary obtuse parallel lines parallelogram perpendicular plane position principle problem produced proof properties PROPOSITION proved radiant reason rectangle rectilineal figure remaining respects side AB side AC surfaces THEOR thing third triangle ABC triangles are equal truths unequal vertex wherefore whole
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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.