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 9
... angles shall be equal , each to each , to which the equal AA 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 AA 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
... angles of the one coincide with the other angles of the other , and are equal to them , namely , the angle ABC to the angle DEF , and the angle ACB to the angle DFE . Wherefore , if two triangles & c . Q.E.D. PROPOSITION 5. THEOREM . The ...
... angles of the one coincide with the other angles of the other , and are equal to them , namely , the angle ABC to the angle DEF , and the angle ACB to the angle DFE . Wherefore , if two triangles & c . Q.E.D. PROPOSITION 5. THEOREM . The ...
Page 11
... angle FAG common to the two triangles AFC , AGB ; therefore the base FC is equal to the base GB , and the triangle ... ACB , which are the angles at the base of the triangle ABC . [ Axiom 3 . And it has also been shewn that the angle FBC ...
... angle FAG common to the two triangles AFC , AGB ; therefore the base FC is equal to the base GB , and the triangle ... ACB , which are the angles at the base of the triangle ABC . [ Axiom 3 . And it has also been shewn that the angle FBC ...
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 ...
Page 13
... angle is without the other triangle ; because AC is equal to AD , [ Hypothesis . the angle ACD is equal to the angle ... ACB , produce AC , AD to E , F. Then because AC is equal to AD , in the triangle ACD , [ Hyp . the angles ECD , FDC , on ...
... angle is without the other triangle ; because AC is equal to AD , [ Hypothesis . the angle ACD is equal to the angle ... ACB , produce AC , AD to E , F. Then because AC is equal to AD , in the triangle ACD , [ Hyp . the angles ECD , FDC , on ...
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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.