Definitions
Postulates
Common Notions
Propositions
Prop. 1 (problem): On the given
straight finite straight-line to construct an equilateral triangle.
Prop. 2 (problem): To position at the given point a straight-line equal to the given line.
Prop. 3 (problem): With two given, unequal straight-lines to take away from the larger a straight-line equal to the smaller.
Prop. 4 (theorem): If two triangles have two sides respectively equal to two sides, and the angle contained by the straight lines equal to the angle, then they will have the base equal to the base and the triangle will be equal to the triangle, and the remaining angles will be respectively equal to the remaining angles which the equal sides subtend.
Prop. 5 (theorem): The angles at the base of isosceles triangles are equal to one another, and when the equal sides are extended the angles under the base will be equal to one another.
Prop. 6 (theorem): If two angles of a triangle are equal to one another, then the sides subtending the equal angles will be equal to one another.
Prop. 7 (theorem): On the same straight-line two different straight-lines respectively equal to the same straight-lines will not be constructed at one and another point on the same sides while having the same limits as the initial lines.
Prop. 8 (theorem): If two triangles have the two sides equal to the two sides respectively, but also have the base equal to the base, then they will have the angle that’s enclosed by the equal straight lines equal to the angle.
Prop. 9 (theorem): To bisect the given rectilinear angle.
Prop. 10 (problem): To bisect the given finite straight-line.
Prop. 11 (problem): To draw a straight line at right angles to the given straight-line from the given point on it.
Prop. 12 (problem): To draw a perpendicular straight line to the given infinite straight-line form the given point, which is not one it.
Prop. 13 (theorem): If a straight-line stood against a straight-line makes angles, it will make either two right-angles or equals to two right angles.
Prop. 14 (theorem): If at some line and the point on it two straight-lines, not positioned on the same sides, make successive angles equal to two right-angles, the straight-lines will be on a straight-line with one another.
Prop. 15 (theorem): If two straight-lines cut one another, they make the angles at the vertex equal to one another.
Prop. 16 (theorem): Upon one of the sides of any triangle being extended, the external angle is larger than each of the interior and opposite angles.
Prop. 17 (theorem): The two angles of any triangle, taken in any way, are smaller than two right-angles.
Prop. 18 (theorem): The larger side of any triangle subtends the larger angle
Prop. 19 (theorem): Subtending the larger angle of any triangle is the larger side. Let there be a triangle,
Prop. 20 (theorem): The two sides of any triangle, taken in any way, are larger than the remaining.
Prop. 21 (theorem): If two straight-lines are constructed within a triangle from the limits on one of the sides, the constructed straight-lines will be smaller than the remaining two sides of the triangle and enclose a larger angle.
Prop. 22 (problem): To construct a triangle from three straight-lines which are equal to three, the given straight-lines.
Prop. 23 (problem): To construct at the given straight-line and the point on it an angle equal to the given rectilinear angle.
Prop. 24 (theorem): If two triangles have the two sides respectively equal to the two sides but have the angle enclosed by the equal straight-lines larger than the angle, then they will have the base larger thn the base.
Prop. 25 (theorem): If two triangles have the two sides respectively equal two sides and the base larger than the base, then they will have the angle that’s enclosed by the equal sides larger than the angle.
Prop. 26 (theorem): If two triangles have two angles respectively equal to two angles and one side equal to one side, whether that at the equal angles or that subtending one of the equal angles, it will also have the remaining sides respectively equal to the remaining sides and the remaining angle to the remaining angle.
Prop. 27 (theorem): If a straight-line falling within two straight-lines makes alternate angles equal to one another, the straight-lines will be parallel to one another.
Prop. 28 (theorem): If a straight-line falling within two straight-lines makes the external angle equal to the angle that’s internal and opposite and on the same sides or the angles that are interior and on the same sides equal to two right-angles, the straight-lines will be parallel to one another..
Prop. 29 (theorem): The straight-line falling within parallel straight-lines makes the alternate angles equal to one another and the external angle equal to the angle that’s internal and opposite and on the same sides and the angles that are interior and on the same sides equal to two right-angles..
Prop. 30 (theorem): The parallels to the same straight-line are also parallel to one another.
Prop. 31 (problem): To draw a straight line through the given point parallel to the given line.
Prop. 32 (theorem): Of every triangle, when one of the sides is extended the external angle is equal to the two interior and opposite angles and the three interior angles of the triangle are equal to two right angles.
Prop. 33 (theorem): The opposite sides and angles of parallelogram regions are equal to one another, and the diameters bisect them
Prop. 34 (theorem): Parallelograms that are on the same base and in the same parallels are equal to one another.
Prop. 35 (theorem): Parallelograms that are on the same base and in the same parallels are equal to one another.
Prop. 36 (theorem): Parallelograms that are on equal bases and are in the same parallels are equal to one another.
Prop. 37 (theorem): Triangles that are on the same base and are in the same parallels are equal to one another.
Prop. 38 (theorem): Triangles that are on equal bases and are in the same parallels are equal to one another.
Prop. 39 (theorem): Equal triangles that are on the same bases and are on the same sides are also in the same parallels.
Prop. 40 (theorem):Equal triangles that are on equal bases and are on the same sides are also in the same parallels.
Prop. 41 (theorem): If a parallelogram has a base that’s the same for a triangle and is in the same parallels, the parallelogram is double the triangle.
Prop. 42 (problem): To construct a parallelogram equal to the given triangle in the given rectilinear angle.
Prop. 43 (theorem): For every parallelogram, the complements of the parallelograms about the diameter are equal to one another.
Prop. 44 (problem): To apply along the given straight-line a parallelogram equal to the given triangle in the given rectilinear angle.
Prop. 45 (problem): To construct a parallelogram equal to the given rectilinear-figure in the given rectilinear angle.
Prop. 46 (problem): From the given straight-line, to describe up a square.
Prop. 47 (theorem): In right-angled triangles the square from the side subtending the right angle is equal to the squares from the sides containing the right angle.
Prop. 48 (theorem): If the square from one of the sides of a triangle is equal is equal to squares from the remaining two sides of the triangle, the angle enclosed by the remaining sides of the triangle is right. .