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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 meet 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 segment shew shewn side BC similar square on AC straight line &c straight line drawn tangent THEOREM touches the circle triangle ABC triangle DEF twice the rectangle vertex Wherefore
Page 23 - greater than AB. Wherefore, the greater angle &c. QED PROPOSITION 20. THEOREM. Any two sides of a triangle are together greater than the third side. Let ABC be a triangle: any two sides of it are together greater than the third side; namely, BA, AC greater than
Page 66 - parts may be equal to the square on the other part. Let AB be the given straight line: it is required to divide it into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part. On AB describe the square ABDC;
Page 184 - PROPOSITION 7. THEOREM. If two triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportionals; then, if each of the remaining angles be either less, or not less, than a right angle, or if one of them be a right angle, the triangles
Page 57 - unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. Let the straight line AB be divided into two equal parts at the point C, and into two unequal parts at the point D: the rectangle AD,DB, together with the square
Page 35 - PROPOSITION 32. THEOREM. If a side of any triangle be produced, the exterior angh is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles. Let ABC be a triangle, and let one of its
Page 200 - same ratio to one another that the polygons have; and the polygons are to one another in the duplicate ratio of their homologous sides. Let ABCDE, FGHKL be similar polygons, and let AB be the side homologous to the side FG : the polygons ABODE, FGHKL may be divided into the same number of similar triangles,
Page 17 - and they are adjacent angles. But 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. [Def. 10.
Page 34 - PROPOSITION 30. THEOREM. Straight lines which are parallel to the same straight line are parallel to each other. Let AB, CD be each of them parallel to EF: AB shall be parallel to CD. Let the straight line GHK cut AB, EF, CD. Then, because GHK cuts the parallel straight lines AB, EF,
Page 288 - triangles which have one angle of the one equal to one angle of the other, have to one another the ratio which is compounded of the ratios of their sides. Then VI. 19 is an immediate consequence of this theorem. For let ABC and
Page 12 - Corollary. Hence every equiangular triangle is also equilateral. PROPOSITION 7. THEOREM. On the same base, and on the same side of it, there cannot be two triangles having their sides which are terminated at one extremity of the base equal to one another, and likewise those which are terminated at the other extremity equal to one another.