The Elements of Euclid for the Use of Schools and Colleges: Comprising the First Six Books and Portions of the Eleventh and Twelfth Books : with Notes, an Appendix, and ExercisesMacmillan, 1867 - 400 Seiten |
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Seite 3
... diameter and the part of the circumference cut off by the diameter . 19. A segment of a circle is the figure contained by a straight line and the circumference which it cuts off . 20. Rectilineal figures are those which are contained by ...
... diameter and the part of the circumference cut off by the diameter . 19. A segment of a circle is the figure contained by a straight line and the circumference which it cuts off . 20. Rectilineal figures are those which are contained by ...
Seite 37
... diameter bisects the par- allelogram , that is , divides it into two equal parts . Note . A parallelogram is a four - sided figure of which the opposite sides are parallel ; and a diameter is the straight line joining two of its ...
... diameter bisects the par- allelogram , that is , divides it into two equal parts . Note . A parallelogram is a four - sided figure of which the opposite sides are parallel ; and a diameter is the straight line joining two of its ...
Seite 38
... diameter ; the opposite sides and angles of the figure shall be equal to one another , and the diameter BC shall bi- sect it . Because AB is parallel to CD , and BC meets them , the alternate angles ABC , BCD are equal to one an- other ...
... diameter ; the opposite sides and angles of the figure shall be equal to one another , and the diameter BC shall bi- sect it . Because AB is parallel to CD , and BC meets them , the alternate angles ABC , BCD are equal to one an- other ...
Seite 41
... diameter AB bisects the parallelogram ; [ I. 34 . and the triangle DBC is half of the parallelogram DBCF , because the diameter DC bisects the parallelogram . [ I. 34 . But the halves of equal things are equal . [ Axiom 7 . Therefore ...
... diameter AB bisects the parallelogram ; [ I. 34 . and the triangle DBC is half of the parallelogram DBCF , because the diameter DC bisects the parallelogram . [ I. 34 . But the halves of equal things are equal . [ Axiom 7 . Therefore ...
Seite 43
... diameter AC bisects the parallelogram . [ I. 34 . Therefore the parallelogram ABCD is also double of the triangle EBC . Wherefore , if a parallelogram & c . Q.E.D. PROPOSITION 42. PROBLEM . To describe a parallelogram that shall BOOK I ...
... diameter AC bisects the parallelogram . [ I. 34 . Therefore the parallelogram ABCD is also double of the triangle EBC . Wherefore , if a parallelogram & c . Q.E.D. PROPOSITION 42. PROBLEM . To describe a parallelogram that shall BOOK I ...
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ABCD angle ABC angle ACB angle BAC angle EDF angles equal Axiom base BC BC is equal bisects the angle centre chord circle ABC circle described circumference Construction Corollary describe a circle diameter double draw a straight equal angles equal to F equiangular equilateral equimultiples Euclid Euclid's Elements exterior angle given circle given point given straight line gnomon Hypothesis inscribed intersect isosceles triangle less Let ABC magnitudes middle point multiple opposite angles opposite sides parallelogram perpendicular plane polygon produced proportionals PROPOSITION 13 Q.E.D. PROPOSITION quadrilateral radius rectangle contained rectilineal figure remaining angle rhombus right angles right-angled triangle segment shew shewn side BC square on AC straight line &c straight line drawn tangent THEOREM touches the circle triangle ABC triangle DEF twice the rectangle vertex Wherefore
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Seite 286 - 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 30 - 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...
Seite 34 - IF a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles...
Seite 37 - If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.
Seite 12 - 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.
Seite 302 - Describe a circle which shall pass through a given point and touch a given straight line and a given circle.
Seite 220 - If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Seite 354 - Prove that the square on any straight line drawn from the vertex of an isosceles triangle to the base, is less than the square on a side of the triangle by the rectangle contained by the segments of the base : and conversely.
Seite 104 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
Seite 96 - The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles.