Euclid in Paragraphs: The Elements of Euclid: Containing the First Six Books and the First Twenty-one Propositions of the Eleventh Book ...T. Stevenson, 1845 - 199 σελίδες |
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Σελίδα 56
... Similar segments of a circle are those in which the angles are equal , or which con- tain equal angles . PROPOSITION I. * 10. J. t 11. 1 . PROB . To find the centre of a given circle . Let ABC be the given circle ; it is required to ...
... Similar segments of a circle are those in which the angles are equal , or which con- tain equal angles . PROPOSITION I. * 10. J. t 11. 1 . PROB . To find the centre of a given circle . Let ABC be the given circle ; it is required to ...
Σελίδα 74
... similar to the seg- +11 Def . 3. ment ADB , and that similar segments of circles contain † equal angles ; the angle ACB is equal to the angle ADB , the exterior to the interior , which is impossible . There- fore , there cannot be two ...
... similar to the seg- +11 Def . 3. ment ADB , and that similar segments of circles contain † equal angles ; the angle ACB is equal to the angle ADB , the exterior to the interior , which is impossible . There- fore , there cannot be two ...
Σελίδα 76
... similar to the segment EDF ; and they are upon equal straight lines BC , EF : § 24. 3 . 4 Нур . 3 Ax . But similar segments of circles upon equal straight lines are equal§ to one another ; therefore the segment BAC is equal to the ...
... similar to the segment EDF ; and they are upon equal straight lines BC , EF : § 24. 3 . 4 Нур . 3 Ax . But similar segments of circles upon equal straight lines are equal§ to one another ; therefore the segment BAC is equal to the ...
Σελίδα 138
... was proper to show the true meaning of them when they are so expressed ; especially since they are very frequently made use of by geometers . BOOK VI . DEFINITIONS . I. SIMILAR rectilineal figures are 138 EUCLID'S ELEMENTS .
... was proper to show the true meaning of them when they are so expressed ; especially since they are very frequently made use of by geometers . BOOK VI . DEFINITIONS . I. SIMILAR rectilineal figures are 138 EUCLID'S ELEMENTS .
Σελίδα 139
... SIMILAR rectilineal figures are those which have their se- veral angles equal , each to each , and the sides about the equal angles propor- tionals . II . " Reciprocal figures , viz . triangles and parallelograms , are " such as have ...
... SIMILAR rectilineal figures are those which have their se- veral angles equal , each to each , and the sides about the equal angles propor- tionals . II . " Reciprocal figures , viz . triangles and parallelograms , are " such as have ...
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ABCD adjacent angles angle ABC angle ACB angle BAC angle BCD angle DEF angle EDF angles equal base BC BC is equal bisected centre circle ABC Constr describe diameter double draw equal angles equal to F equiangular equimultiples ex æquali exterior angle fore four magnitudes fourth given circle given straight line gnomon greater ratio join less Let ABC Let the straight likewise meet multiple opposite angle parallel to BC parallelogram perpendicular plane polygon proportionals Q. E. D. PROPOSITION rectangle contained remaining angle right angles segment semicircle sides square of AC straight line AB straight line AC THEOR THEOR.-If three straight lines tiple touches the circle triangle ABC triangle DEF twice the rectangle wherefore
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Σελίδα 12 - If two triangles have two sides of the one equal to two sides of the...
Σελίδα 29 - THE straight lines which join the extremities of two equal and parallel straight lines, towards the same parts, are also themselves equal and parallel.
Σελίδα 143 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Σελίδα 69 - If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle ; the angles which this line makes with the line touching the circle, shall be equal to the angles which are in the alternate segments of the circle.
Σελίδα 20 - If from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle. Let the two straight lines BD.
Σελίδα 102 - A LESS magnitude is said to be a part of a greater magnitude, when the less measures the greater; that is, ' when the less is contained a certain number of times
Σελίδα 44 - If a straight line be divided into any two parts, the squares of the whole line and of one of the parts are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts at the point C : the squares of AB, BC shall be equal to twice the rectangle AB, BC, together with the square of AC.
Σελίδα 71 - The opposite angles of any quadrilateral figure described in a circle, are toe/ether equal to two right angles. Let ABCD be a quadrilateral figure in the circle ABCD : any two of its opposite angles are together equal to two right angles.
Σελίδα 66 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...
Σελίδα 83 - If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.