Euclid's Elements of plane geometry [book 1-6] with explanatory appendix, and supplementary propositions, by W.D. Cooley1840 |
From inside the book
Results 1-5 of 21
Page 24
... remaining angles opposite to equal sides are respectively equal ( A to D and B to E ) : and the triangles are equal in every respect . C F Let one of the triangles , as ACB , be so applied to the other , DFE , that the point C may ...
... remaining angles opposite to equal sides are respectively equal ( A to D and B to E ) : and the triangles are equal in every respect . C F Let one of the triangles , as ACB , be so applied to the other , DFE , that the point C may ...
Page 36
... remaining sides and angle of the one triangle be respectively equal to the remaining sides and angle of the other . First , let the given equal sides be BC and EF , which lie between the equal angles ; then the sides subtending the ...
... remaining sides and angle of the one triangle be respectively equal to the remaining sides and angle of the other . First , let the given equal sides be BC and EF , which lie between the equal angles ; then the sides subtending the ...
Page 37
... remaining sides and angles equal ( Prop . 4 ) , viz . AC = DF and / BAC = / EDF . B E G F Secondly , let the given equal sides be sides subtending equal angles , as AC and DF ; now in this case , the sides CB , FE lying between the ...
... remaining sides and angles equal ( Prop . 4 ) , viz . AC = DF and / BAC = / EDF . B E G F Secondly , let the given equal sides be sides subtending equal angles , as AC and DF ; now in this case , the sides CB , FE lying between the ...
Page 40
... remaining angles also equal . COR . 2. - If one angle of a triangle be a right angle , the other two are together equal to a right angle . COR . 3. - In an equilateral triangle , each angle is a third of two right angles , and is ...
... remaining angles also equal . COR . 2. - If one angle of a triangle be a right angle , the other two are together equal to a right angle . COR . 3. - In an equilateral triangle , each angle is a third of two right angles , and is ...
Page 41
... remaining sides and angles respectively equal ( Prop . 26 ) ; that is to say , AB = CD , and AC = BD ; but these are the opposite sides of the parallelogram : and also if to the equal angles ADC , DAB , be added the equals BDA , CAD ...
... remaining sides and angles respectively equal ( Prop . 26 ) ; that is to say , AB = CD , and AC = BD ; but these are the opposite sides of the parallelogram : and also if to the equal angles ADC , DAB , be added the equals BDA , CAD ...
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.