Elements of geometry, based on Euclid, books i-iii1876 - 119 pages |
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Page 12
... third sides , equal ; and the two triangles shall be equal , and their other angles shall be equal , each to each , viz . , those to which the equal sides are opposite . Or , If two sides and the contained angle of one triangle be re ...
... third sides , equal ; and the two triangles shall be equal , and their other angles shall be equal , each to each , viz . , those to which the equal sides are opposite . Or , If two sides and the contained angle of one triangle be re ...
Page 26
... third side . Let ABC be a triangle ; Any two sides of it are together greater than the third side . CONSTRUCTION . - Produce BA to the point D , making AD equal to AC ( I. 3 ) , and join DC . PROOF . - Because DA is equal to AC , the ...
... third side . Let ABC be a triangle ; Any two sides of it are together greater than the third side . CONSTRUCTION . - Produce BA to the point D , making AD equal to AC ( I. 3 ) , and join DC . PROOF . - Because DA is equal to AC , the ...
Page 27
... third side ( I. 20 ) , the two sides BA , AE , of the triangle BAE are greater than BE . To each of these add EC . Therefore the sides BA , AC , are B greater than BE , EC ( Ax . 4 ) . A D E ... BA + AC BC . BA + AC > BE + Again ...
... third side ( I. 20 ) , the two sides BA , AE , of the triangle BAE are greater than BE . To each of these add EC . Therefore the sides BA , AC , are B greater than BE , EC ( Ax . 4 ) . A D E ... BA + AC BC . BA + AC > BE + Again ...
Page 28
... third ( I. 20 ) . Let A , B , C be the three given straight lines , of which any two whatever are greater than the third - namely , A and B greater than C , A and C greater than B , and B and C greater than A ; It is required to make a ...
... third ( I. 20 ) . Let A , B , C be the three given straight lines , of which any two whatever are greater than the third - namely , A and B greater than C , A and C greater than B , and B and C greater than A ; It is required to make a ...
Page 31
... third angle of the one equal to the third angle of the other . Or , If two angles and a side in one triangle be respectively equal to two angles and a corresponding side in another triangle , the triangles shall be equal in every ...
... third angle of the one equal to the third angle of the other . Or , If two angles and a side in one triangle be respectively equal to two angles and a corresponding side in another triangle , the triangles shall be equal in every ...
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Common terms and phrases
AB is equal AC and CD adjacent angles alternate angles angle ABC angle ACB angle BAC angle BCD angle contained angle DEF angle EDF angle equal angles BGH angles CBA base BC BC is equal bisect centre circle ABC circumference diagonal diameter double draw equal circles equal to AC equal to twice EUCLID'S ELEMENTS exterior angle given circle given point given rectilineal angle given straight line gnomon greater interior and opposite isosceles triangle less Let ABC Let the straight opposite angles parallel to BC parallelogram perpendicular produced PROOF PROOF.-Because Q.E.D. Proposition rectangle AD rectangle AE rectangle contained remaining angle right angle Const right angles Ax segment semicircle side BC square described square on AC touches the circle triangle ABC triangle DEF twice the rectangle
Popular passages
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 equal to two right angles.
Page 13 - 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. Let ABC be an isosceles triangle, of which the side AB is equal to AC, and let the straight lines AB, AC be produced to D and E.
Page 7 - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
Page 17 - If two triangles have two sides of the one equal to two sides of the...
Page 53 - If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it ; the angle contained by these two sides is a right angle.
Page 9 - If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...
Page 71 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.
Page 9 - Things which are double of the same, are equal to one another. 7. Things which are halves of the same, are equal to one another.
Page 34 - Wherefore, if a straight line, &c. QED PROPOSITION XXVIII. THEOREM. If a straight line falling upon two other straight lines, make the exterior angle equal to the interior and opposite upon the same side of the line ; or make the interior angles upon the same side together equal to two right angles ; the two straight lines shall be parallel to one another.
Page 69 - If a straight line be bisected, and produced to any point; the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line, which is made up of the half and the part produced.