Elements of plane geometry, book i, containing nearly the same propositions as the first book of Euclid's Elements |
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Seite 44
Let BAC be a given rectilineal angle , it is required to bisect it , that is , to divide into two equal parts . ... and on DE describe ( 10 ) the isosceles triangle DEF ; join AF ; the straight line AF bisects the E angle BAC .
Let BAC be a given rectilineal angle , it is required to bisect it , that is , to divide into two equal parts . ... and on DE describe ( 10 ) the isosceles triangle DEF ; join AF ; the straight line AF bisects the E angle BAC .
Seite 45
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 ... Let AB be a given finite straight line , it is required to bisect it .
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 ... Let AB be a given finite straight line , it is required to bisect it .
Seite 46
Bisect ( 15 ) AC in the point E ; join BE F and produce it to F , and make EF equal to EB ; join FC , and produce AC to G. Because the side AE is equal EC and BE to EF , and the included angles AEB and D CEF are equal , because ( 7 ) ...
Bisect ( 15 ) AC in the point E ; join BE F and produce it to F , and make EF equal to EB ; join FC , and produce AC to G. Because the side AE is equal EC and BE to EF , and the included angles AEB and D CEF are equal , because ( 7 ) ...
Seite 50
3 ) the circle EGF , meeting AB in F and G , bisect ( 15 ) FG in AF H , and join CF , CH , CG , the straight line CH drawn from the point C is perpendicular to AB . in the triangles FCH and GCH the side FH is equal to HG , CF to CG ...
3 ) the circle EGF , meeting AB in F and G , bisect ( 15 ) FG in AF H , and join CF , CH , CG , the straight line CH drawn from the point C is perpendicular to AB . in the triangles FCH and GCH the side FH is equal to HG , CF to CG ...
Seite 51
The perpendicular drawn from the middle of the base of an isosceles triangle passes through the vertex and bisects the vertical angle . PROPOSITION XXV . THEOR . В. Of all the straight lines that can be drawn from the same point to a ...
The perpendicular drawn from the middle of the base of an isosceles triangle passes through the vertex and bisects the vertical angle . PROPOSITION XXV . THEOR . В. Of all the straight lines that can be drawn from the same point to a ...
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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.