Elements of Geometry, Containing the First Six Books of Euclid |
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Seite 3
A rhomboid has its opposite sides and angles equal to one another , but all its sides are not equal , nor its angles right angles . 33. All other four sided figures besides these are called trapeziums . 34. Parallel right lines are ...
A rhomboid has its opposite sides and angles equal to one another , but all its sides are not equal , nor its angles right angles . 33. All other four sided figures besides these are called trapeziums . 34. Parallel right lines are ...
Seite 7
one triangle equal to the other triangle ; also the remain- ing angles of the one shall be equal to the remaining angles of the other , each to each , which are opposite to the equal sides . Let there be two triangles , ABC , DEF ...
one triangle equal to the other triangle ; also the remain- ing angles of the one shall be equal to the remaining angles of the other , each to each , which are opposite to the equal sides . Let there be two triangles , ABC , DEF ...
Seite 8
Whence the triangle BFC is equal to the triangle CGB ; and the remaining angles equal to the remaining angles , each to each , to which the equal sides are opposite . Therefore the angle FBC is equal to the angle GCB ; and the angle BCF ...
Whence the triangle BFC is equal to the triangle CGB ; and the remaining angles equal to the remaining angles , each to each , to which the equal sides are opposite . Therefore the angle FBC is equal to the angle GCB ; and the angle BCF ...
Seite 17
One side of any triangle being produced , the exterior angle is greater than either of the interior and opposite angles . A F a 15. 1 . E b 4. 1 . D Let ABC be a triangle , and let one of its sides BC be produced to D. The exterior ...
One side of any triangle being produced , the exterior angle is greater than either of the interior and opposite angles . A F a 15. 1 . E b 4. 1 . D Let ABC be a triangle , and let one of its sides BC be produced to D. The exterior ...
Seite 18
There- B -D of the triangle ACB is greater than the interior and opposite angle ABC ; add ACB , which is common . fore the angles ACD , ACB , are greater than the angles ABC , ACB . But ACD , ACB , are equal to two right angles , whence ...
There- B -D of the triangle ACB is greater than the interior and opposite angle ABC ; add ACB , which is common . fore the angles ACD , ACB , are greater than the angles ABC , ACB . But ACD , ACB , are equal to two right angles , whence ...
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ABC is equal ABCD alternate angle ABC angle ACB angle BAC angles equal applied base base BC bisected centre circle ABC circumference common consequently demonstrated described diameter divided double draw equal equal angles equiangular equilateral equimultiples exceed extreme fall figure fore four fourth Geometry given circle given rectilineal given right line greater half hence inscribed join less Let ABC magnitudes manner mean meet multiple opposite parallel parallelogram pass perpendicular PROBLEM produced proportional PROPOSITION ratio reason rectangle rectangle contained rectilineal figure remaining angle right angles right line AC sector segment shown sides similar square taken THEOREM third triangle ABC whence Wherefore whole
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Seite xxiv - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. XIX. "A segment of a circle is the figure contained by a straight line, and the circumference it cuts off.
Seite 72 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...
Seite 31 - The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.
Seite 146 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Seite 25 - And because the angle ABC is equal to the angle BCD, and the angle CBD to the angle ACB, therefore the whole angle ABD is equal to the whole angle ACD • (ax.
Seite 6 - To bisect a given finite straight line, that is, to divide it into two equal parts. Let AB be the given straight line : it is required to divide it intotwo equal parts.
Seite 71 - DH; (I. def. 15.) therefore DH is greater than DG, the less than the greater, which is impossible : therefore no straight line can be drawn from the point A, between AE and the circumference, which does not cut the circle : or, which amounts to the same thing, however great an acute angle a straight line makes with the diameter at the point A, or however small an angle it makes with AE, the circumference must pass between that straight line and the perpendicular AE.
Seite 97 - To describe a square about a given circle. Let ABCD be the given circle ; it is required to describe a square about it. . Draw two diameters AC, BD of the circle ABCD, at right angles to one another, and through the points A, B, • 17.3. C, D, draw...
Seite 5 - ... equal to them, of the other. Let ABC, DEF be two triangles which have the two sides AB, AC equal to the two sides DE, DF, each to each, viz. AB equal to DE, and AC to DF ; but the base CB greater than the base EF ; the angle BAC is likewise greater than the angle EDF.
Seite 86 - From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle.