The Elements of Euclid for the Use of Schools and Colleges: Comprising the First Six Books and Portions of the Eleventh and Twelfth Books

Macmillan and Company, Limited, 1882 - 400 páginas

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Página 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
Página 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;
Página 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
Página 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
Página 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
Página 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,
Página 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.
Página 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,
Página 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
Página 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.

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