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PROPOSITIONS BOOK I
Proposition 1. On a given finite straight line to construct an equilateral triangle.
Proposition 2. To place at a given point (as an extremity) a straight line equal to a given straight line.
Proposition 3.
Given two unequal straight lines, to cut off from the greater a straight
line equal to the less.
Proposition 4. If two triangles have the two sides equal to two sides respectively,
and have the angles contained by the equal straight lines equal,
they will also have the base equal to the base, the triangle will be equal to the triangle,
and the remaining angles will be equal to the remaining angles respectively,
namely those which the equal sides subtend.
Proposition 5. In isosceles triangles the angles at the base are equal to one another,
and, if the equal straight lines be produced further, the angles under the base
will be equal to one another.
Proposition 6. If in a triangle two angles be equal to one another,
the sides which subtend the equal angles will also be equal to one another.
Proposition
7. Given two straight lines constructed on a straight
line (from its extremities) and meeting in a point, there cannot be constructed on the
same straight line (from its extremities), and on the same side of it,
two other straight lines meeting in another point and equal to the former two respectively,
namely each to that which has the same extremity with it.
Proposition
8. If two triangles have the two sides equal to two sides respectively,
and have also the base equal to the base, they will also have the angles equal
which are contained by the equal straight lines.
Proposition
9. To bisect a given rectilineal angle.
Proposition
10. To bisect a given finite straight line.
Proposition
11. To draw a straight line at right angles to a given straight line from a given point on it.
Proposition
12. To a given infinite straight line, from a given point which is not on it,
to draw a perpendicular straight line.
Proposition
13. If a straight line set up on a straight line make angles,
it will make either two right angles or angles equal to two right angles.
Proposition
14. If with any straight line, and at a point on it, two straight lines not lying
on the same side make the adjacent angles equal to two right angles,
the two straight lines will be in a straight line with one another.
Proposition
15. If two straight lines cut one another, they make the vertical angles equal to one another.
Porism: If two straight lines cut one another, they will make the angles at the point of section equal to four right angles.
Proposition
16. In any triangle, if one of the sides be produced, the exterior angle is greater
than either of the interior and opposite angles.
Proposition
17. In any triangle two angles taken together in any manner are less than two right angles.
Proposition
18. In any triangle the greater side subtends the greater angle.
Proposition
19. In any triangle the greater angle is subtended by the greater side.
Proposition
20. In any triangle two sides taken together in any manner are greater than the remaining one.
Proposition
21. If on one of the sides of a triangle, from its extremities,
there be constructed two straight lines meeting within the triangle,
the straight lines so constructed will be less than the remaining two sides of the triangle,
but will contain a greater angle.
Proposition
22. Out of three straight lines, which are equal to three given straight lines,
to construct a triangle: thus it is necessary that two of the straight lines taken together
in any manner should be greater than the remaining one.
Proposition
23. On a given straight line and at a point on it to construct
a rectilineal angle equal to a given rectilineal angle.
Proposition
24. If two triangles have the two sides equal to two sides respectively,
but have the one of the angles contained by the equal straight lines greater than the other,
they will also have the base greater than the base.
Proposition
25. If two triangles have the two sides equal to two sides respectively,
but have the base greater than the base, they will also have the one of the angles contained
by the equal straight lines greater than the other.
Proposition
26. If two triangles have the two angles equal to two angles respectively,
and one side equal to one side, namely, either the side adjoining the equal angles,
or that subtending one of the equal angles, they will also have the remaining sides equal to
the remaining sides and the remaining angle to the remaining angle.
Proposition
27. If a straight line falling on two straight lines make the alternate angles equal to one another,
the straight lines will be parallel to one another.
Proposition
28. If a straight line falling on two straight lines make the exterior angle equal
to the interior and opposite angle on the same side, or the interior angles
on the same side equal to two right angles, the straight lines will be parallel to one another.
Proposition
29. A straight line falling on parallel straight lines makes the alternate angles equal
to one another, the exterior angle equal to the interior and opposite angle,
and the interior angles on the same side equal to two right angles.
Proposition
30. Straight lines parallel to the same straight line are also parallel to one another.
Proposition
31. Through a given point to draw a straight line parallel to a given straight line.
Proposition
32. In any triangle, if one of the sides be produced, the exterior angle is
equal to the two interior and opposite angles, and the three interior angles
of the triangle are equal to two right angles.
Proposition
33. The straight lines joining equal and parallel straight lines
(at the extremities which are) in the same directions (respectively)
are themselves also equal and parallel.
Proposition
34. In parallelogrammic areas the opposite sides and angles are equal
to one another, and the diameter bisects the areas.
Proposition
35. Parallelograms which are on the same base and in the same parallels are equal to one another.
Proposition
36. Parallelograms which are on equal bases and in the same parallels are equal to one another.
Proposition
37. Triangles which are on the same base and in the same parallels are equal to one another.
Proposition
38. Triangles which are on equal bases and in the same parallels are equal to one another.
Proposition
39. Equal triangles which are on the same base and on the same side are also in the same parallels.
Proposition
40. Equal triangles which are on equal bases and on the same side are also in the same parallels.
Proposition
41. If a parallelogram have the same base with a triangle and be in the same parallels,
the parallelogram is double of the triangle.
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