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 12
... join DC . Then , because in the triangles DBC , ACB , DB is equal to AC , and BC is common to both , B [ I. 3 . [ Construction . the two sides DB , BC are equal to the two sides AC , CB , each to each ; and the angle DBC is equal to the ...
... join DC . Then , because in the triangles DBC , ACB , DB is equal to AC , and BC is common to both , B [ I. 3 . [ Construction . the two sides DB , BC are equal to the two sides AC , CB , each to each ; and the angle DBC is equal to the ...
Page 15
... join DE , and on DE , on the side remote from A , describe the equi- lateral triangle DEF . [ I. 1 . B Join AF . The straight line AF shall bisect the angle BAC . Because AD is equal to AE , [ Construction . and AF is common to the two ...
... join DE , and on DE , on the side remote from A , describe the equi- lateral triangle DEF . [ I. 1 . B Join AF . The straight line AF shall bisect the angle BAC . Because AD is equal to AE , [ Construction . and AF is common to the two ...
Page 16
... join CF. [ I. 1 . The straight line CF drawn from the given point C shall be at right angles to the given straight line AB . Because DC is equal to CE , [ Construction . and CF is common to the two triangles DCF , ECF ; the two sides DC ...
... join CF. [ I. 1 . The straight line CF drawn from the given point C shall be at right angles to the given straight line AB . Because DC is equal to CE , [ Construction . and CF is common to the two triangles DCF , ECF ; the two sides DC ...
Page 17
... join CH . [ I. 10 . E H B D The straight line CH drawn from the given point C shall be perpendicular to the given straight line AB . Join CF , CG . Because FH is equal to HG , [ Construction . and HC is common to the two triangles FHC ...
... join CH . [ I. 10 . E H B D The straight line CH drawn from the given point C shall be perpendicular to the given straight line AB . Join CF , CG . Because FH is equal to HG , [ Construction . and HC is common to the two triangles FHC ...
Page 20
... join BE and produce it to F , making EF equal to EB , [ I. 3 . and join FC . Because AE is equal to EC , and BE to EF ; [ Constr . the two sides AE , EB are equal to the two sides CE , EF , each to each ; and the angle AEB is equal to ...
... join BE and produce it to F , making EF equal to EB , [ I. 3 . and join FC . Because AE is equal to EC , and BE to EF ; [ Constr . the two sides AE , EB are equal to the two sides CE , EF , each to each ; and the angle AEB is equal to ...
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Common terms and phrases
ABCD angle ABC angle ACB angle BAC angle EDF angles equal Axiom base BC BC is equal 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 radius 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 286 - 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 30 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...
Page 34 - IF a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles...
Page 37 - 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 together equal to two right angles.
Page 12 - 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 302 - Describe a circle which shall pass through a given point and touch a given straight line and a given circle.
Page 220 - If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Page 354 - 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.
Page 104 - 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 96 - The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles.