The Elements of Euclid for the Use of Schools and Colleges: Comprising the First Six Books and Portions of the Eleventh and Twelfth Books : with Notes, an Appendix, and Exercises |
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Page 2
... angle is the inclination of two straight lines to one another , which meet together , but are not in the same ... ABC , or CBA ; the angle which is contained by the straight lines AB , DB is named the angle ABD , or DBA ; and the angle ...
... angle is the inclination of two straight lines to one another , which meet together , but are not in the same ... ABC , or CBA ; the angle which is contained by the straight lines AB , DB is named the angle ABD , or DBA ; and the angle ...
Page 9
... angles shall be equal , each to each , to which the equal B sides are opposite , namely , the angle ABC to the angle DEF , and the angle ACB to the angle DFE . For if the triangle ABC be applied to the triangle BOOK I. 3 , 4 . 9.
... angles shall be equal , each to each , to which the equal B sides are opposite , namely , the angle ABC to the angle DEF , and the angle ACB to the angle DFE . For if the triangle ABC be applied to the triangle BOOK I. 3 , 4 . 9.
Page 10
... ABC be applied to the triangle DEF , so that the point A may be on the point D , and the straight line AB on the ... angle BAC is equal to the angle EDF . B AA [ Hypothesis . Therefore also the point C will coincide with the point F ...
... ABC be applied to the triangle DEF , so that the point A may be on the point D , and the straight line AB on the ... angle BAC is equal to the angle EDF . B AA [ Hypothesis . Therefore also the point C will coincide with the point F ...
Page 11
... angle FAG common to the two triangles AFC , AGB ; therefore the base FC is equal to the base GB , and the triangle ... ABC is equal to the remain- ing angle ACB , which are the angles at the base of the triangle ABC . [ Axiom 3 . And it ...
... angle FAG common to the two triangles AFC , AGB ; therefore the base FC is equal to the base GB , and the triangle ... ABC is equal to the remain- ing angle ACB , which are the angles at the base of the triangle ABC . [ Axiom 3 . And it ...
Page 12
... angles of a triangle be equal to one another , the sides also which subtend , or are oppo- site to , the equal angles , shall be equal to one another . Let ABC be a triangle , having the angle ABC equal to the angle ACB : the side AC ...
... angles of a triangle be equal to one another , the sides also which subtend , or are oppo- site to , the equal angles , shall be equal to one another . Let ABC be a triangle , having the angle ABC equal to the angle ACB : the side AC ...
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Common terms and phrases
ABCD AC is equal angle ABC angle ACB angle BAC angle EDF angles equal Axiom base BC bisects the angle centre chord circle ABC circle described circumference Construction Corollary describe a circle diameter double draw a straight equal angles equal to F equiangular equilateral equimultiples Euclid Euclid's Elements exterior angle given circle given point given straight line gnomon Hypothesis inscribed intersect isosceles triangle less Let ABC magnitudes middle point multiple opposite angles opposite sides parallelogram perpendicular plane polygon produced proportionals PROPOSITION 13 Q.E.D. PROPOSITION quadrilateral rectangle contained rectilineal figure remaining angle rhombus right angles right-angled triangle segment shew shewn side BC square on AC straight line &c straight line drawn tangent THEOREM touches the circle triangle ABC triangle DEF twice the rectangle vertex Wherefore
Popular passages
Page 35 - IF a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles ; and the three interior angles of every triangle are equal to two right angles.
Page 67 - In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle, is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle between the perpendicular and the obtuse angle, Let ABC be an obtuse-angled triangle, having the obtuse angle...
Page 73 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a Right Angle; and the straight line which stands on the other is called a Perpendicular to it.
Page 284 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Page 50 - In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.
Page 57 - If a straight line be divided into two equal parts, and also into two unequal parts ; the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line.
Page 10 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.
Page 227 - If two straight lines be at right angles to the same plane, they shall be parallel to one another. Let the straight lines AB, CD be at right angles to the same plane : AB is parallel to CD.
Page 102 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
Page 352 - Prove that the square on any straight line drawn from the vertex of an isosceles triangle to the base, is less than the square on a side of the triangle by the rectangle contained by the segments of the base : and conversely.