Euclid of Alexandria

A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.

And the point is called the center of the circle.

A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a line also bisects the circle.

A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.

Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

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 three acute angles.

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 all other quadrilaterals besides these be called trapezia.

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

To draw a straight line from any point to any point.

To produce a finite straight line continuously in a straight line.

To describe a circle with any center and radius.

That all right angles are equal to one another.

That, if a straight line falling on two straight lines makes 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.

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.

If in a triangle two angles be equal to one another, the sides also which subtend the equal angles will be equal to one another.

The sum of the angles in any triangle is equal to two right angles.

In any right-angled triangle the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

If a straight line be cut in extreme and mean ratio, the greater segment squared is equal to the rectangle contained by the whole and the lesser segment.