## The Elements of Euclid: With Many Additional Propositions, & Explanatory Notes, Etc, Part 1 |

### From inside the book

Results 1-5 of 17

Page 5

...

...

**diagonal**. 26. A PARALLELOGRAM is a quadrilateral figure whose opposite sides are parallel . SCHOLIUM . If a**diagonal**AC be drawn to any parallelogram ABCD , and lines GH and EF be drawn respectively parallel to two contiguous sides of ... Page 35

...

...

**diagonal**BC . DEMONSTRATION . Because in the triangles ABC and DBC the sides AB and CD are equal ( a ) , the side BC common to both , and the angle ABC equal to the alternate angle BCD ( ) , therefore the two sides AC and BD are equal ... Page 36

...

...

**diagonal**( CB ) . DEMONSTRATION . Because AB is parallel to CD ( a ) , the angle ABC is equal to the alternate angle BCD ( 6 ) ; and because AC is parallel to BD ( a ) , the angle ACB is equal to the alternate angle DBC ( 6 ) . Then ... Page 37

...

...

**diagonals**of a parallelogram bisect each other . For since , in the triangles AEC and DEB , the side AC is equal to the side BD ( a ) , the angle EAC to EDB , and the angle ECA to EBD ( b ) , therefore the side AE is equal to ED , and ... Page 39

...

...

**diagonals**BA and CD bisect the equal parallelograms ( c ) , the triangle ABC is half the parallelogram EBCA , and the W ( a ) I. 31 ( b ) I. 35 . ( c ) I. 34 . ( d ) Ax . 7 . triangle DBC is half the parallelogram DBCF ; therefore the ...### Other editions - View all

### Common terms and phrases

AC and CB AC is equal angle ABC angle BCD angle equal area to double area to twice bisect chord circle ABC circumference Constr CONSTRUCTION COROLLARY DB is equal DEMONSTRATION diagonal divided double the rectangle draw equal angles equal in area equal to AC equilateral Euclid external angle Find the center finite straight line Geometry given angle given line greater Hypoth HYPOTHESES intersect join less line BC lines be drawn magnitude major premiss opposite sides parallel parallelogram perpendicular predicate premises produced proposition quadratic equation reductio ad absurdum right angles SCHOLIA SCHOLIUM second power segment sides AC square on AC square on half squares on AB syllogism termed THEOREM THEOREM.-If triangle ABC twice the rectangle twice the square vertex whole line

### Popular passages

Page 24 - 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 114 - In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side...

Page xiv - A circle is a plane figure contained by one line, which is called the circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference are equal to one another : 16. And this point is called the centre of the circle.

Page 13 - The difference between any two sides of a triangle is less than the third side.

Page 111 - AFC. (in. 21.) Hence in the triangles ADE, AFC, there are two angles in the one respectively equal to two angles in the other, consequently, the third angle CAF is equal to the third angle DAB ; therefore the arc DB is equal to the arc CF, (in.

Page 89 - ... the centre of the circle shall be in that line. Let the straight line DE touch the circle ABC in C, and from C let CA be drawn at right angles to DE ; the centre of the circle is in CA.

Page 70 - EQUAL circles are those of which the diameters are equal, or from the centres of which the straight lines to the circumferences are equal. ' This is not a definition, but a theorem, the truth of ' which is evident; for, if the circles be applied to one ' another, so that their centres coincide, the circles ' must likewise coincide, since the straight lines from

Page 34 - To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle.

Page 22 - If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...