## Elements of Plane Geometry According to Euclid |

### From inside the book

Page 4

**An**oblong or rectangle is that which has all its angles right angles , but has not all its**sides**equal . ... and which , being produced ever**so**far both ways , do not meet . POSTULATES .**1**. Let it be granted that a**straight line may**be ... Page 9

... and the other angles , to which the equal

... and the other angles , to which the equal

**sides**are opposite , shall be equal ,**each**to**each**; namely , the angle ...**so**that the point A**may**be on D , and the**straight line**AB**upon**DE , the point B shall coincide with the point E ... Page 25

If a

If a

**straight line**falling**upon**two other**straight lines**makes the alternate angles equal to**one**another , these two ... D. In**like**manner it**may**be demonstrated , that they do not meet towards A , Č ; but those**straight lines**which ... Page 35

Therefore the base AB is not greater than DE , and in a

Therefore the base AB is not greater than DE , and in a

**similar**manner it**may**be proved not to be less ... be**upon**the same**sides**of their bases , or that their bases be in the same**straight line**; for when the two figures do not fulfil ... Page 43

In a

In a

**given straight line**to find a point equally distant from two**given**points . 11. ... If two lines bisect perpendicularly two**sides**of a triangle , the perpendicular , from their point of intersection**upon**the base , will bisect it .### What people are saying - Write a review

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### Common terms and phrases

ABCD AC is equal alternate angle ABC angle ACB angle BAC base bisected called centre chord circle circle ABC circumference common contained demonstrated described diameter difference divided double draw equal equal angles equiangular equilateral equimultiples extremities figure fore former four fourth geometry given straight line greater half inscribed join less Let ABC magnitudes manner mean measure meet multiple namely parallel parallelogram pass perpendicular polygon principles PROBLEM produced proportional PROPOSITION proved radius ratio reason rectangle rectangle contained rectilineal figure remaining angle respectively right angles segment shown sides similar square square of AC Take taken tangent THEOREM third triangle ABC twice vertical wherefore whole

### Popular passages

Page 1 - Things which are equal to the same thing are equal to one another. 2. If equals be added to equals the wholes are equal. 3. If equals be taken from equals the remainders are equal. 4. If equals be added to unequals the wholes are unequal. 5. If equals be taken from unequals the remainders are unequal. 6. Things which are double of the same thing are equal to one another.

Page 73 - 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 9 - To bisect a given finite straight line, that is, to divide it into two equal parts. Let AB be the given straight line : it is required to divide it intotwo equal parts.

Page 4 - If two triangles have two sides of the one equal to two sides of the...

Page 139 - Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides. Let AC, CF be equiangular parallelograms, having the angle BCD equal to the angle ECG ; the ratio of the parallelogram AC to the parallelogram CF, is the same with the ratio which is compounded of the ratios of their sides. Let BC, CG, be placed in a straight line ; therefore DC and CE are also in a straight line (2.

Page 23 - Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Page 129 - 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 80 - A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure about which it is described. 7. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle.

Page 27 - Parallelograms upon equal bases, and between the same parallels, are equal to one another.

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.