## Elements of Geometry: Containing the First Six Books of Euclid : with a Supplement on the Quadrature of the Circle, and the Geometry of Solids : to which are Added Elements of Plane and Spherical Trigonometry |

### From inside the book

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**ABC**is an**equilateral**triangle . Because the point A is the cen- tre of the circle BCD , AC is**equal**מן C A B E ( 11. Definition ) to AB ; and because the point B is the centre of the cir - ` cle ACE , BC is**equal**to AB : But it has ... Page 13

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**equal**to one another ; therefore the straight line AL is**equal**to BC . Wherefore , from the given point A , a ...**ABC**, DEF be two triangles which have the two sides AB , AC**equal**to the two sides DE , DF , each to each , viz . AB to DE ... Page 14

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**equal**to it . Therefore also the whole triangle**ABC**shall coincide with the whole triangle DEF , so that the spaces which they contain or their areas are**equal**; and the remaining angles of the one shall coincide with the remaining ... Page 15

... equal to one another , the sides which subtend , or are opposite to them , are also equal to one another . Let ABC be a triangle having the angle

... equal to one another , the sides which subtend , or are opposite to them , are also equal to one another . Let ABC be a triangle having the angle

**ABC equal**to the angle ACB ; the side AB is also equal to the side AC . For , if AB be not ... Page 16

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**equal**to AD in the triangle ACD , the angles ECD , FDC upon the other side of the base CD are**equal**( 5. 1. ) to one ...**ABC**, DEF be two triangles having the two sides AB , AC ,**equal**to the two sides DE , DF , each to each , viz . AB ...### Other editions - View all

### Common terms and phrases

ABC is equal ABCD adjacent angles altitude angle ABC angle ACB angle BAC base BC bisected centre chord circle ABC circumference cosine cylinder demonstrated described diameter divided draw equal and similar equal angles equiangular equilateral equilateral polygon equimultiples Euclid exterior angle fore four right angles given rectilineal given straight line greater Hence hypotenuse inscribed join less Let ABC Let the straight magnitudes meet multiple opposite angle parallel parallelogram parallelopiped perpendicular polygon prism PROB PROP proportional proposition quadrilateral radius ratio rectangle contained rectilineal figure remaining angle right angled triangle SCHOLIUM segment semicircle shewn side BC sine solid angle spherical angle spherical triangle square straight line BC THEOR touches the circle triangle ABC triangle DEF wherefore

### Popular passages

Page 51 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.

Page 12 - AB; but things which are equal to the same are equal to one another...

Page 80 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

Page 288 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Page 35 - Parallelograms upon the same base and between the same parallels, are equal to one another.

Page 81 - If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles made by this line with the line which touches the circle, shall be equal to the angles in the alternate segments of the circle.

Page 52 - 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 in the point C ; the squares of AB, BC are equal to twice the rectangle AB, BC, together with the square of AC.

Page 127 - 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 23 - 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.

Page 19 - The angles which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles. Let the straight line AB make with CD, upon one side of it the angles CBA, ABD ; these are either two right angles, or are together equal to two right angles. For, if the angle CBA be equal to ABD, each of them is a right angle (Def.