Elements of geometry, based on Euclid, books i-iii1876 - 119 pages |
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Page 9
... coincide with one another , that is , which exactly fill the same space , are equal to one another . 9. The whole is greater than its part . 10. Two straight lines cannot inclose a space . 11. All right angles are equal to one another ...
... coincide with one another , that is , which exactly fill the same space , are equal to one another . 9. The whole is greater than its part . 10. Two straight lines cannot inclose a space . 11. All right angles are equal to one another ...
Page 13
... coincide with the point E , because AB is equal to DE ( Hypothesis ) . And AB coinciding with DE , AC shall coincide with DF , because the angle BAC is equal to the angle EDF ( Hyp . ) . Therefore also the point C shall coincide with ...
... coincide with the point E , because AB is equal to DE ( Hypothesis ) . And AB coinciding with DE , AC shall coincide with DF , because the angle BAC is equal to the angle EDF ( Hyp . ) . Therefore also the point C shall coincide with ...
Page 17
... coincides with the base EF , AC = DF , and BC EF . Make BC coincide with EF . B .. BA , AC respective- But the sides BA , PROPOSITIONS . 17.
... coincides with the base EF , AC = DF , and BC EF . Make BC coincide with EF . B .. BA , AC respective- But the sides BA , PROPOSITIONS . 17.
Page 18
... coincides with the base EF , the ly coincide sides BA , AC must coincide with the sides ED , DF . with ED , DF . Make AE = AD , A DEF e- quilateral . .. DAF LEAF . Therefore the angle BAC coincides with the angle EDF , and is equal to ...
... coincides with the base EF , the ly coincide sides BA , AC must coincide with the sides ED , DF . with ED , DF . Make AE = AD , A DEF e- quilateral . .. DAF LEAF . Therefore the angle BAC coincides with the angle EDF , and is equal to ...
Page 77
... coincide with the base . 16. The square on any straight line drawn from the vertex of an isosceles triangle , together with the rectangle contained by the segments of the base , is equal to the square upon a side of the triangle . 17 ...
... coincide with the base . 16. The square on any straight line drawn from the vertex of an isosceles triangle , together with the rectangle contained by the segments of the base , is equal to the square upon a side of the triangle . 17 ...
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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.