Elements of geometry, based on Euclid, books i-iii1876 - 119 pages |
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Page 44
... passing through A can be parallel to BC , except AD ; Therefore AD is parallel to BC . Therefore , equal triangles , & c . Q. E. D. Proposition 40. - Theorem . Equal triangles upon the same side of equal bases , that are in the same ...
... passing through A can be parallel to BC , except AD ; Therefore AD is parallel to BC . Therefore , equal triangles , & c . Q. E. D. Proposition 40. - Theorem . Equal triangles upon the same side of equal bases , that are in the same ...
Page 45
... passing through A , can be parallel to BF , except AD ; Therefore AD is parallel to BF . Therefore , equal triangles , & c . Proposition 41. - Theorem . If a parallelogram and a triangle be upon the same base , and between the same ...
... passing through A , can be parallel to BF , except AD ; Therefore AD is parallel to BF . Therefore , equal triangles , & c . Proposition 41. - Theorem . If a parallelogram and a triangle be upon the same base , and between the same ...
Page 47
... passes ; and BK , KD the other parallelograms , which make up the whole figure ABCD , and are therefore called the complements . The complement BK shall be equal to the complement KD . PROOF . - Because ABCD is a parallelogram , and AC ...
... passes ; and BK , KD the other parallelograms , which make up the whole figure ABCD , and are therefore called the complements . The complement BK shall be equal to the complement KD . PROOF . - Because ABCD is a parallelogram , and AC ...
Page 55
... pass through the same point . 22. The difference of any two sides of a triangle is less than the third side . 23. If the angles at the base of a right - angled isosceles triangle be bisected , the bisecting line includes an angle which ...
... pass through the same point . 22. The difference of any two sides of a triangle is less than the third side . 23. If the angles at the base of a right - angled isosceles triangle be bisected , the bisecting line includes an angle which ...
Page 85
... passes through the centre ; of those which fall on the concave circum- ference , the greatest is that which passes through the centre , and of the rest , that which is nearer to the one passing through the centre is always greater than ...
... passes through the centre ; of those which fall on the concave circum- ference , the greatest is that which passes through the centre , and of the rest , that which is nearer to the one passing through the centre is always greater than ...
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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.