Euclid

To construct a regular tetrahedron and comprehend it in a sphere.

To construct a regular octahedron and comprehend it in a sphere.

To construct a regular cube and comprehend it in a sphere.

To construct a regular icosahedron and comprehend it in a sphere.

To construct a regular dodecahedron and comprehend it in a sphere.

There are no other figures, besides the five aforesaid, which are constructed by equilateral and equiangular figures equal to one another.

If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts together with the square on the straight line between the points of section is equal to the square on the half.

If a straight line be cut into two equal parts, and to one of them a straight line be added, the square on the whole with the added straight line and the square on the added straight line are together double of the square on the half and of the square on the straight line made up of the half and the added straight line.

The ends of a line are points.

The edges of a surface are lines.

Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

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.

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.

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.

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.

To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.

In any triangle two angles taken together in any manner are less than two right angles.