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 xv
... propositions . The propositions are of two kinds , pro- blems and theorems . In a problem something is required to be done ; in a theorem some new principle is asserted to be true . A proposition consists of various parts . We have ...
... propositions . The propositions are of two kinds , pro- blems and theorems . In a problem something is required to be done ; in a theorem some new principle is asserted to be true . A proposition consists of various parts . We have ...
Seite xvi
... proposition itself . The word hypothesis is used in the same sense as supposition . To assist the student in following the steps of the reasoning , references are given to the results already ob- tained which are required in the ...
... proposition itself . The word hypothesis is used in the same sense as supposition . To assist the student in following the steps of the reasoning , references are given to the results already ob- tained which are required in the ...
Seite 7
... PROPOSITION 1. PROBLEM . To describe an equilateral triangle on a given finite straight line . Let AB be the given straight line : it is required to describe an equilateral triangle on AB . A B E From the centre A , at the distance AB ...
... PROPOSITION 1. PROBLEM . To describe an equilateral triangle on a given finite straight line . Let AB be the given straight line : it is required to describe an equilateral triangle on AB . A B E From the centre A , at the distance AB ...
Seite 12
... PROPOSITION 6. THEOREM . If two angles of a triangle be equal to one another , the sides also which subtend , or are oppo- site to , the equal angles , shall be equal to one another . Let ABC be a triangle , having the angle ABC equal ...
... PROPOSITION 6. THEOREM . If two angles of a triangle be equal to one another , the sides also which subtend , or are oppo- site to , the equal angles , shall be equal to one another . Let ABC be a triangle , having the angle ABC equal ...
Seite 15
... PROPOSITION 10. PROBLEM . 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 into two equal parts . Describe on it an equilateral triangle ...
... PROPOSITION 10. PROBLEM . 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 into two equal parts . Describe on it an equilateral triangle ...
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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.