Euclid

Rectilineal figures are those which are contained by straight lines.

Trilateral figures are those contained by three straight lines, quadrilateral figures those contained by four straight lines, and multilateral figures those contained by more than four straight lines.

Among 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.

Among 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.

Let it be granted that from any point to any point a straight line may be drawn.

Let it be granted that a terminated straight line may be produced to any length in a straight line.

And 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.

The angles at the base of an isosceles triangle are equal to one another.

If two triangles have 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 subtend the equal sides.

In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are equal to two right angles.

If a straight line be cut into two equal parts and 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 it be added a straight line in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line.

In any right-angled triangle, the square on the side opposite the right angle is equal to the sum of the squares on the other two sides.

To draw a straight line perpendicular to a given straight line from a given point not on it.

To construct a square on a given straight line.

There are infinitely many prime numbers.

If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.

If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

There are no royal roads to geometry.