## Euclid in Paragraphs: The Elements of Euclid: Containing the First Six Books and the First Twenty-one Propositions of the Eleventh Book ... |

### From inside the book

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**BC is equal**to BA : But it has been proved that CA is equal to AB ; there- fore CA , CB are each of them equal to AB ; But things which are equal to the same are equal to one tion . another ; therefore CA is equal to CB ; wherefore CA ... Page 8

... equal to BC . B G K H D A E Because the point B is the centre of the circle CGH ,

... equal to BC . B G K H D A E Because the point B is the centre of the circle CGH ,

**BC is equal**to BG ; * And because D is the centre of the circle GKL , DL is equal to DG , and † DA , DB , parts of them , are equal ; therefore the ... Page 9

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**equal**to two sides of the other , each to each ; and have likewise the angles contained by those sides**equal**to one ...**BC**shall be**equal**to the base EF ; and the triangle ABC to the triangle DEF ; and the other angles to which B CE the ... Page 10

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**equal**, and the triangles be**equal**, and their other angles to which the**equal**sides are opposite shall be**equal**...**BC**is common to the two triangles BFC , CGB ; wherefore the triangles are**equal**† , and their remaining angles , each to ... Page 11

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**equal**to AC , one of them is greater than the other . Let AB be the greater ; and from it cut * off DB**equal**to AC , the less , and join DC ; therefore , because in the triangles DBC , ACB , DB is**equal**to AC , † and**BC**common to both ...### Other editions - View all

### Common terms and phrases

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

### Popular passages

Page 12 - If two triangles have two sides of the one equal to two sides of the...

Page 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.

Page 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.

Page 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.

Page 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.

Page 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

Page 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.

Page 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.

Page 66 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...

Page 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.