^^Euclide. Book1 propositions.
[Fundamentals of Plane Geometry: Straight-Lines,
Trisides.]
- To construct an equilateral triangle on a given finite straight line.
>>>
c: corollario: any line segment is part of a triangle.
- To place a straight line equal to a given straight line with one end at
a given point. >>>
- To cut off from the greater of two given unequal straight lines a
straight line equal to the less.
- If two triangles have two sides equal to two sides respectively, and
have the angles contained by the equal straight lines equal, then they also
have the base equal to the base, the triangle equals the triangle, and the
remaining angles equal the remaining angles respectively, namely those
opposite the equal sides. >>>
- In isosceles triangles the angles at the base equal one another, and, if
the equal straight lines are produced further, then the angles under the
base equal one another.
- If in a triangle two angles equal one another, then the sides opposite
the equal angles also equal one another.
- Given two straight lines constructed from the ends of a straight line
and meeting in a point, there cannot be constructed from the ends of the
same straight line, and on the same side of it, two other straight lines
meeting in another point and equal to the former two respectively, namely
each equal to that from the same end.
- If two triangles have the two sides equal to two sides respectively, and
also have the base equal to the base, then they also have the angles equal
which are contained by the equal straight lines.
- To bisect a given rectilinear angle.
- To bisect a given finite straight line.
- To draw a straight line at right angles to a given straight line from a
given point on it.
- To draw a straight line perpendicular to a given infinite straight line
from a given point not on it.
- If a straight line stands on a straight line, then it makes either two
right angles or angles whose sum equals two right angles.
- If with any straight line, and at a point on it, two straight lines not
lying on the same side make the sum of the adjacent angles equal to two
right angles, then the two straight lines are in a straight line with one
another.
- If two straight lines cut one another, then they make the vertical
angles equal to one another.
Corollary. If two straight lines cut one another, then they will make the
angles at the point of section equal to four right angles.
- In any triangle, if one of the sides is produced, then the exterior
angle is greater than either of the interior and opposite angles.
- In any triangle the sum of any two angles is less than two right angles.
- In any triangle the angle opposite the greater side is greater.
- In any triangle the side opposite the greater angle is greater.
- In any triangle the sum of any two sides is greater than the remaining
one.
- If from the ends of one of the sides of a triangle two straight lines
are constructed meeting within the triangle, then the sum of the straight
lines so constructed is less than the sum of the remaining two sides of the
triangle, but the constructed straight lines contain a greater angle than
the angle contained by the remaining two sides.
- To construct a triangle out of three straight lines which equal three
given straight lines: thus it is necessary that the sum of any two of the
straight lines should be greater than the remaining one.
- To construct a rectilinear angle equal to a given rectilinear angle on a
given straight line and at a point on it.
- If two triangles have two sides equal to two sides respectively, but
have one of the angles contained by the equal straight lines greater than
the other, then they also have the base greater than the base.
- If two triangles have two sides equal to two sides respectively, but
have the base greater than the base, then they also have the one of the
angles contained by the equal straight lines greater than the other.
- If two triangles have two angles equal to two angles respectively, and
one side equal to one side, namely, either the side adjoining the equal
angles, or that opposite one of the equal angles, then the remaining sides
equal the remaining sides and the remaining angle equals the remaining
angle.
- If a straight line falling on two straight lines makes the alternate
angles equal to one another, then the straight lines are parallel to one
another.
- If a straight line falling on two straight lines makes the exterior
angle equal to the interior and opposite angle on the same side, or the sum
of the interior angles on the same side equal to two right angles, then the
straight lines are parallel to one another.
- 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 sum of the interior angles on the same side equal to
two right angles.
- Straight lines parallel to the same straight line are also parallel to
one another.
- To draw a straight line through a given point parallel to a given
straight line.
- In any triangle, if one of the sides is produced, then the exterior
angle equals the sum of the two interior and opposite angles, and the sum of
the three interior angles of the triangle equals two right angles.
- Straight lines which join the ends of equal and parallel straight lines
in the same directions are themselves equal and parallel.
- In parallelogrammic areas the opposite sides and angles equal one
another, and the diameter bisects the areas.
- Parallelograms which are on the same base and in the same parallels
equal one another.
- Parallelograms which are on equal bases and in the same parallels equal
one another.
- Triangles which are on the same base and in the same parallels equal one
another.
- Triangles which are on equal bases and in the same parallels equal one
another.
- Equal triangles which are on the same base and on the same side are also
in the same parallels.
- Equal triangles which are on equal bases and on the same side are also
in the same parallels.
- If a parallelogram has the same base with a triangle and is in the same
parallels, then the parallelogram is double the triangle.
- To construct a parallelogram equal to a given triangle in a given
rectilinear angle.
- In any parallelogram the complements of the parallelograms about the
diameter equal one another.
- To a given straight line in a given rectilinear angle, to apply a
parallelogram equal to a given triangle.
- To construct a parallelogram equal to a given rectilinear figure in a
given rectilinear angle.
- To describe a square on a given straight line.
- In right-angled triangles the square on the side opposite the right
angle equals the sum of the squares on the sides containing the right angle.
- If in a triangle the square on one of the sides equals the sum of the
squares on the remaining two sides of the triangle, then the angle contained
by the remaining two sides of the triangle is right.
credits: ©2002 David E. Joyce
Links
Book1 map.