## Euclid for beginners, books i. and ii., with simple exercises by F.B. Harvey |

### From inside the book

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**ABC**is an equilateral**triangle**described upon AB . PROOF . - Because A is the centre of the circle BCD , therefore AB AC ( def . 15 ) . Similarly , because B is the centre of the circle ACE , therefore BA = BC . But it has been proved ... Page 4

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**triangles ABC**and DEF let the sides AB and AC , and their angle BAC , in the former the sides DE and DF , and their angle EDF , in the latter , each to each . Then it is to be proved that 1. The base BC 2. The**triangle ABC**3. The angle ABC ... Page 5

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**triangle ABC**: the triangle DEF . 3. The angle ABC = the angle DEF . 4. The angle ACB = the angle DFE . Wherefore , If two triangles , & c . Q. E. D. Exercises . 1. Given the straight lines AB and CD , of which AB is the greater ; it is ... Page 6

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**ABC**be an isosceles**triangle**with the side AB = the side AC , and let AB and AC be produced to D and E respectively . Then it is to be proved that 1. The angles**ABC**and ACB , at the base other . each 2. The angles DBC and ECB , upon ... Page 7

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**ABC**= the remaining angle ACB ( ax . 3 ) , and these are the angles at the base . Therefore , it is proved , as ...**triangle**, & ' c . Q. E. D. Cor . Every equilateral**triangle**is also equiangular . NOTE . It may assist the scholar ...### Other editions - View all

### Common terms and phrases

ABC and ABD AC and CD adjacent angles alternate angle angle ABC angle ACB angle AGH angle BAC angle CEB angle DEF angle EDF angle GHD Arithmetic BA and AC base BC Beginners bisected CONSTRUCTION.-1 crown 8vo Dictionary double the square draw Edition English Grammar English History equilateral Euclid exterior angle Gallic War Geography given straight line gnomon greater Greek half a right i.e. the angle interior and opposite join Latin Let ABC line be divided LONGMANS Manual note 2 def opposite angle parallel parallelogram post 8vo produced PROOF.-Because Proposition proved Q. E. D. Exercise Q. E. D. PROP rectangle contained rectilineal figure right angles School side AB side AC small 8vo square on AC Stepping-Stone straight line CD THEOREM triangle ABC twice the rect twice the rectangle vols Wherefore

### Popular passages

Page 48 - 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 88 - 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 on the line between the points of section, is equal to the square on half the line.

Page 14 - To draw a straight line at right angles to a given straight line, from a given point in the same.

Page 36 - If two triangles have two angles of the one equal to two angles of the other, each to each, and also one side of the one equal to the corresponding side of the other, the triangles are congruent.

Page 64 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Page 108 - In every triangle, the square on the side subtending an acute angle, is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle, and the acute angle. Let ABC be any triangle, and the angle at B an acute angle; and on BC one of the sides containing it, let fall the perpendicular...

Page 47 - To draw a straight line through a given point parallel to a given straight line. Let A be the given point, and BC the given straight line.

Page 104 - 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 52 - 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 20 - If, at a point in a straight line, two other straight lines upon the opposite sides of it, make the adjacent angles, together equal to two right angles, these two straight lines shall be in one and the same straight line.