Euclid's Elements of plane geometry [book 1-6] with explanatory appendix, and supplementary propositions, by W.D. Cooley1840 |
From inside the book
Results 6-10 of 30
Page 116
... proportional to the sides BA , AC , which contain the angle ad- B jacent to the external bisected angle . D ECA = / CAD For , if CE be drawn parallel to DA , ( i . Prop . 29 ) = DAF ( Hyp . ) = / CEA ( i . Prop . 29 ) ; and therefore AE ...
... proportional to the sides BA , AC , which contain the angle ad- B jacent to the external bisected angle . D ECA = / CAD For , if CE be drawn parallel to DA , ( i . Prop . 29 ) = DAF ( Hyp . ) = / CEA ( i . Prop . 29 ) ; and therefore AE ...
Page 117
... proportional , the triangles shall be equiangular , and have their equal angles opposite to homologous sides . From the extremities of EF , one of the sides of the tri- angle DEF , draw EG and FG , making E D F B FEG equal to ABC , and ...
... proportional , the triangles shall be equiangular , and have their equal angles opposite to homologous sides . From the extremities of EF , one of the sides of the tri- angle DEF , draw EG and FG , making E D F B FEG equal to ABC , and ...
Page 118
... proportional . First , let it be assumed that the angles ABC , DEF are each less than a right angle : then , if it be sup- posed that the angles ACB , DFE contained by the proportional sides , are not equal , let ZACB be the greater ...
... proportional . First , let it be assumed that the angles ABC , DEF are each less than a right angle : then , if it be sup- posed that the angles ACB , DFE contained by the proportional sides , are not equal , let ZACB be the greater ...
Page 119
... proportional ( vi . Prop . 4 ) , and are therefore similar ( vi . Def . 1 ) . În like manner it may be proved that the triangle ADC is similar to BAC ; but ADB has been shown to be similar to BAC ; therefore ADB and ADC are similar to ...
... proportional ( vi . Prop . 4 ) , and are therefore similar ( vi . Def . 1 ) . În like manner it may be proved that the triangle ADC is similar to BAC ; but ADB has been shown to be similar to BAC ; therefore ADB and ADC are similar to ...
Page 120
... proportional to those of a given divided straight line ( AC ) . Let AB , AC be so placed as to contain any angle , and join BC ; and from D , E , the points in which AC is divided , draw DF , EG parallel to BC . The given line AB is ...
... proportional to those of a given divided straight line ( AC ) . Let AB , AC be so placed as to contain any angle , and join BC ; and from D , E , the points in which AC is divided , draw DF , EG parallel to BC . The given line AB is ...
Common terms and phrases
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
Popular passages
Page 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'.
Page 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.
Page 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.
Page 129 - ... figures are to one another in the duplicate ratio of their homologous sides.
Page 47 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.
Page 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.
Page 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.
Page 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.
Page 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.
Page 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.