Euclid's Elements of plane geometry [book 1-6] with explanatory appendix, and supplementary propositions, by W.D. Cooley1840 |
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Página 59
... inscribed in a circle , and terminated both ways in the circumference . 3. A straight line is said to touch a circle when it meets , and being produced , does not cut the circle . 4. Circles are said to touch one another which meet but ...
... inscribed in a circle , and terminated both ways in the circumference . 3. A straight line is said to touch a circle when it meets , and being produced , does not cut the circle . 4. Circles are said to touch one another which meet but ...
Página 68
... inscribed in a circle are equally distant from the centre ; and also , straight lines equally distant from the centre are equal . From C , the centre of the circle BAED , draw CF per- pendicular to AB , and CG perpendicular to ED , and ...
... inscribed in a circle are equally distant from the centre ; and also , straight lines equally distant from the centre are equal . From C , the centre of the circle BAED , draw CF per- pendicular to AB , and CG perpendicular to ED , and ...
Página 69
... inscribed in the circle . K Again , since FE is nearer to the centre than HG , the per- pendicular CD is less than CK ( iii . Def . 6 ) ; and , as CDF is a right angle , CF2 = CD2 + DF2 ( i . Prop . 47 ) ; and in like manner CH2 = CK2 + ...
... inscribed in the circle . K Again , since FE is nearer to the centre than HG , the per- pendicular CD is less than CK ( iii . Def . 6 ) ; and , as CDF is a right angle , CF2 = CD2 + DF2 ( i . Prop . 47 ) ; and in like manner CH2 = CK2 + ...
Página 74
... inscribed in a circle , are together equal to two right angles . Draw the diagonals AD , BC ; and then because angles in the same segment are equal , △ ACB = / ADB , and △ BCD = △ BAD ; + therefore △ ACB BCD , that is to say , ACD ...
... inscribed in a circle , are together equal to two right angles . Draw the diagonals AD , BC ; and then because angles in the same segment are equal , △ ACB = / ADB , and △ BCD = △ BAD ; + therefore △ ACB BCD , that is to say , ACD ...
Página 80
... inscribed in a circle , s ACB , ADB are together equal to two right angles ( iii . Prop . 22 ) ; but s BAE , BAF are also equal to two right angles ( i . Prop . 13 ) , and therefore BAE + BAF = / ACD + LADB ; from these equals take away ...
... inscribed in a circle , s ACB , ADB are together equal to two right angles ( iii . Prop . 22 ) ; but s BAE , BAF are also equal to two right angles ( i . Prop . 13 ) , and therefore BAE + BAF = / ACD + LADB ; from these equals take away ...
Términos y frases comunes
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
Pasajes populares
Página 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'.
Página 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.
Página 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.
Página 129 - ... figures are to one another in the duplicate ratio of their homologous sides.
Página 47 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.
Página 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.
Página 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.
Página 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.
Página 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.
Página 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.