Euclid's Elements of plane geometry [book 1-6] with explanatory appendix, and supplementary propositions, by W.D. Cooley1840 |
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Seite 20
... angle . 24. An obtuse - angled triangle is that which has an obtuse angle . 25. An acute - angled triangle is that ... external angle ; and the angle of the triangle having the same vertex with it , is the internal adjacent angle . The ...
... angle . 24. An obtuse - angled triangle is that which has an obtuse angle . 25. An acute - angled triangle is that ... external angle ; and the angle of the triangle having the same vertex with it , is the internal adjacent angle . The ...
Seite 25
... external angles at the base are also equal . Produce the equal sides AB , AC ... angle A common to both triangles , is contained by the equal sides ; the ... external angles , are equal to each other . COROLLARY . - Hence it follows that ...
... external angles at the base are also equal . Produce the equal sides AB , AC ... angle A common to both triangles , is contained by the equal sides ; the ... external angles , are equal to each other . COROLLARY . - Hence it follows that ...
Seite 31
... angles . In like manner it may be proved that the vertically opposite angles AFD and BFC are equal . PROP . XVI . THEOR . If a side of a triangle ( BAC ) is produced , the external angle is greater than either of the internal remote angles ...
... angles . In like manner it may be proved that the vertically opposite angles AFD and BFC are equal . PROP . XVI . THEOR . If a side of a triangle ( BAC ) is produced , the external angle is greater than either of the internal remote angles ...
Seite 32
... angles . A Produce the side BC to D ; and because △ ACD is an external angle , it is greater than the internal remote angle CAB ( Prop . 16 ) ; add to both BCA , and then DCA +2 BCA will be greater than / BCA + CAB ( Ax . 4 ) ; but ...
... angles . A Produce the side BC to D ; and because △ ACD is an external angle , it is greater than the internal remote angle CAB ( Prop . 16 ) ; add to both BCA , and then DCA +2 BCA will be greater than / BCA + CAB ( Ax . 4 ) ; but ...
Seite 37
... exterior to , and therefore also greater than △ DEF ( Prop . 16 ) , which is absurd . Therefore the sides CB , FE are not unequal ; and since they are equal , and also AC = DF , and ≤ ACB = / DFE , the tri- angles ... angle AEF is greater ...
... exterior to , and therefore also greater than △ DEF ( Prop . 16 ) , which is absurd . Therefore the sides CB , FE are not unequal ; and since they are equal , and also AC = DF , and ≤ ACB = / DFE , the tri- angles ... angle AEF is greater ...
Häufige Begriffe und Wortgruppen
ACDB adjacent angles angles equal antecedent Axioms base bisected centre chord circumference coincide consequently Const definition demonstrated describe diagonal diameter difference divided draw equal angles equal Prop equal sides equiangular equilateral triangle equimultiples Euclid Euclid's Elements external angle extremities fore fourth fractional Geometry given angle given circle given line given point given straight line given triangle greater hypotenuse inscribed internal intersect isosceles triangle less line drawn lines be drawn magnitudes manner meeting multiple opposite angles parallel parallelogram perpendicular point of contact PROB produced proportional Proposition quadrilateral figure rectangle contained rectilinear figure remaining angles respectively equal right angle segment semiperimeter sides AC sides equal square of half subtending taken tangent THEOR third triangles ABC unequal vertex whole line
Beliebte Passagen
Seite 126 - The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. D c A' D' Hyp. In triangles ABC and A'B'C', ZA = ZA'. To prove AABC = ABxAC. A A'B'C' A'B'xA'C' Proof. Draw the altitudes BD and B'D'.
Seite 155 - If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles are congruent.
Seite 83 - If a straight line touch a circle, and from the point of contact a chord be drawn, the angles which this chord makes with the tangent are equal to the angles in the alternate segments.
Seite 129 - ... figures are to one another in the duplicate ratio of their homologous sides.
Seite 47 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.
Seite 90 - BFE : (i. def. 10.) therefore, in the two triangles, EAF, EBF, there are two angles in the one equal to two angles in the other, each to each ; and the side EF, which is opposite to one of the equal angles in each, is common to both ; therefore the other sides are equal ; (i.
Seite 117 - A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment as the greater segment is to the less.
Seite 56 - If a straight line be bisected, and produced to any point, the square of the whole line thus produced, and the square of the part of it produced, are together double of the square of half the line bisected, and of the square of the line made up of the half and the part produced.
Seite 60 - Iff a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.
Seite 78 - Upon the same straight line, and upon the same side of it, there cannot be two similar segments of circles, not coinciding with one another. If it be possible. let the two similar segments of circles, viz. ACB' ADB be upon the same side of the same straight line AB, not coinciding with one another.