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
... ARCH or arc of a circle is any part of the circum- ference . 8. A SEGMENT of a circle is the figure contained by a straight line , and the arch which it cuts off . 1D 9. An angle in a segment is the angle made [59] ...
... ARCH or arc of a circle is any part of the circum- ference . 8. A SEGMENT of a circle is the figure contained by a straight line , and the arch which it cuts off . 1D 9. An angle in a segment is the angle made [59] ...
Página 60
... arch intercepted between the straight lines which make the angle . 11. A SECTOR of a circle is the figure contained by two rays and the arch between them . 12. Similar segments of circles are those which contain equal angles . PROP . I ...
... arch intercepted between the straight lines which make the angle . 11. A SECTOR of a circle is the figure contained by two rays and the arch between them . 12. Similar segments of circles are those which contain equal angles . PROP . I ...
Página 72
... arch ( AB ) . D First , suppose that C , the centre of the circle , is in AD , one of the legs of the angle at the circum- ference . In this case it is manifest that Z ACB is equal to s CDB , CBD together ( i . Prop . 32 ) ; but these ...
... arch ( AB ) . D First , suppose that C , the centre of the circle , is in AD , one of the legs of the angle at the circum- ference . In this case it is manifest that Z ACB is equal to s CDB , CBD together ( i . Prop . 32 ) ; but these ...
Página 73
... arch . PROP . XXI . THEOR . All angles ( BAD , BED ) in the same segment ( BAED ) of a circle are equal to one another . First , let the segment BAED be greater than a semicircle : from C the centre of the circle draw CB , CD ; then ...
... arch . PROP . XXI . THEOR . All angles ( BAD , BED ) in the same segment ( BAED ) of a circle are equal to one another . First , let the segment BAED be greater than a semicircle : from C the centre of the circle draw CB , CD ; then ...
Página 75
... arches AEB , CFD may lie at the same side ; then because AB - CD , the point B must fall on D , and the line AB must wholly coincide with CD ; and the similar seg- ments AEB , CFD , being then upon the same straight line and at the same ...
... arches AEB , CFD may lie at the same side ; then because AB - CD , the point B must fall on D , and the line AB must wholly coincide with CD ; and the similar seg- ments AEB , CFD , being then upon the same straight line and at the same ...
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.