This document contains definitions and proofs from Euclid's Elements, one of the most influential mathematics texts ever written. It details fundamental concepts in geometry such as constructing equilateral triangles, bisecting angles and lines, drawing perpendicular lines, properties of parallel lines, and theorems about triangles and their angles and sides. The document explores the building blocks of geometric proofs and relationships established by Euclid over two millennia ago.

1. Euclid’s
Elements

2. Euclid’s
Elements

3. Euclid’s
Elements

8. Thousands of years after its
author died, here we are still
marveling at this text…
Take a moment to appreciate the
nuances of this poem of a
mathematics book.

9. By hovering your mouse over a
prop, you can see both

10. By hovering your mouse over a
prop, you can see both
- the props that went into its proof

11. By hovering your mouse over a
prop, you can see both
-the props that went into its proof
and
- the later props that rely on it.

12. Happy
geometring!
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13. 9/27/2011 13

14. On a given finite straight line -
- to construct an equilateral triangle.
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15. At a given point, with a given
straight line -
- to place (as an extremity) a straight line
equal to the given straight line.
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16. Given two unequal straight lines -
- to cut off from the greater a straight line
equal to the less.
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17. If two triangles each have two of their
respective sides and the contained
angles equal to each other -
- the triangle will be equal to the triangle;
- the remaining angles will be equal to the
remaining angles, respectively.
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18. In isosceles triangles -
- the angles at the base will be equal to each
other;
- as will be the angles under the base.
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19. 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.
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20. If two straight lines (constructed
at the extremities of a straight
line) meet at a point -
- there cannot be constructed, on the same
side of the line, two lines equal to the other
straight lines which meet at a different point.
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21. If two triangles have the two
sides and the base equal,
respectively -
- the angles contained by those straight lines
will also be equal.
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22. Given a rectilineal angle -
- to bisect it.
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23. Given a finite straight line -
- to bisect it.
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24. To a given straight line, and
from a given point on it -
- to draw a straight line at right angles.
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25. To a given straight line, from a
given point which is not on it -
- to draw a perpendicular straight line.
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26. If a straight line set up on a
straight line makes angles -
- it will make either two right angles, or
angles equal to two right angles.
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27. If two straight lines, meeting another
straight line at the same point, make the
adjacent angles equal to two right angles -
- the two straight lines will be in a straight
line with each other.
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28. If two straight lines cut one
another -
-they make the vertical angles equal to
one another.
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29. If one of the sides of any
triangle be produced -
- the resulting exterior angle is greater than
either of the interior, opposite angles.
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30. Two angles of any triangle,
when taken together -
- are less than two right angles.
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31. The greater side in
any triangle -
- subtends the greater angle.
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32. The greater angle in
any triangle -
- is subtended by the greater side.
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33. Two sides of any triangle, when
taken together in any manner,
- are greater than the remaining one.
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34. If, from the extremities of one side of a
triangle, two straight lines meeting within
the triangle be constructed -
- the straight lines will be less than the
triangle’s remaining two sides;
- but they will contain a greater angle.
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35. Given three straight lines (as long as
two taken together are greater than
the remaining long) -
- to construct a triangle using three straight
lines equal to those given.
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36. Given a rectilineal angle, as well as
a given point on a straight line -
- to construct at that point another,
equal rectilineal angle.
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37. If two triangles have two of their sides
respectively equal, but one of the
contained angles is larger than the other -
- the triangle with the larger angle will
also have a larger base.
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38. If two triangles have two of their
sides respectively equal, but one of
the bases is larger than the other -
- the triangle with the larger base will also
have a larger angle.
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39. If two triangles have two of their
angles respectively equal, as well as any
one of their respective sides equal -
-the remaining respective sides will be equal;
-as will be the remaining angle.
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40. 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.
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41. If a straight line falling on two straight lines make
(a)the exterior angle equal to the interior, opposite
angle on the same side, or (b)the interior angles on
the same side equal to two right angles -
- the straight lines will be parallel to
one another.
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42. If a straight line falls on parallel
straight lines -
- alternate angles are equal;
- the exterior angle is equal to the interior, opposite angle;
- and interior angles on the same side are equal to two right
angles.
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43. Straight lines parallel to the
same straight line -
- are also parallel to one another.
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44. Given a straight line and a point
(not on the line) -
- to draw through the point a line
parallel to the one given.
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45. If one of the sides of any
triangle be produced -
-the resulting exterior angle is equal to the
two interior, opposite angles;
- and the triangle’s three interior angles are
equal to two right angles.
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46. Straight lines that join equal
and parallel straight lines, in the
same respective directions -
-are themselves equal and parallel
as well.
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47. In parallelogrammic
areas -
-opposite sides are equal to one another;
- opposite angles are equal to one another;
-and the diameter bisects the areas.
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48. Parallelograms which share a
base and are in the same
parallels -
- are equal to each other.
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49. Parallelograms which are on
equal (but not shared bases) and
in the same parallels -
- are equal to one another.
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50. Triangles which are share a base
and are in the same parallels -
- are equal to one another.
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51. Triangles which are on equal
(not shared) bases and in the
same parallels -
- are equal to one another.
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52. If equal triangles share a base
and are on the same side -
- they are also in the same
parallels.
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53. If equal triangles be on equal
(not shared bases) and on the
same side -
- they are also in the same
parallels.
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54. If a parallelogram share a base
with a triangle and be in the
same parallels -
- the parallelogram is double
of the triangle
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55. From a given triangle, in a given
rectilineal angle -
- to construct a parallelogram
equal to the triangle.
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56. In any parallelogram -
- the compliments about the diameter
are equal to one another.
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57. From a given triangle, within a
given rectilineal angle, and to a
given straight line -
- to apply a parallelogram equal to
the triangle.
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58. Given any rectilineal figure and
a rectilineal angle -
- to construct in the angle a
parallelogram equal to the figure.
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59. On a given straight line -
- to describe a square.
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60. In right-angled triangles -
- the square on the side subtending
the right angle is equal to the squares on
the sides containing the right angle.
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61. If the square on one of a triangle’s
sides be equal to the squares on its
remaining two sides -
- the angle contained by the remaining
two sides of the triangle is right.
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