Go to A small guide to the following pages/Book II
Book I
Definitions, postulates and common notions.
Proposition 1. Given a straight line, to construct on it an equilateral triangle.
Proposition 2. Given a point and a straight line, to draw from the point a straight line of equal length to the given straight line.
Proposition 3. Given two unequal straight lines, to cut off from the greater a straight line of equal length to the less.
Proposition 4. (Side-Angle-Side, SAS) If two triangles have a common angle and if the legs of that angle are common too, then the remaining side, the remaining angles, and the areas of the triangles will be common.
Proposition 5. (A well known fact of isosceles triangles) In isosceles triangles the angles at the base are equal to one another, and if the equal sides are produced further, the angles under the base will be equal to one another.
Proposition 6. (The converse of part of proposition 5).
Proposition 7. .
Proposition 8. (Side-Side-Side, SSS).
Proposition 9. .
Proposition 10. Given a finite straight line, to bisect it.
Proposition 11. .
Proposition 12. Given an infinite straight line and a point (that is not on the line), it is possible to draw a perpendicular straight line from the point to the line.
Proposition 13. .
Proposition 14. .
Proposition 15. .
Proposition 16. .
Proposition 17. In any triangle the sum of two angles is less than 180 degrees.
Proposition 18. In any triangle, the greater side is opposite the greater angle.
Proposition 19. In any triangle, the greater angle is opposite the greater side.
Proposition 20. (The triangle inequality). In any triangle, the sum of two sides is greater than the remaining side.
Proposition 21. In a triangle, let two straight lines be drawn from the endpoints of one side such that they meet inside the triangle. The sum of the length of these lines will be less than the sum of the remaining sides of the triangle.
Proposition 22. How to construct a triangle from three given straight lines (requirement: the sum of any two of the lines must exceed the third line's length, see proposition 20).
Proposition 47. The theorem of Pythagoras. In right-angled triangles the square of the hypotenuse is equal to the sum of the squares of the two legs.
Proposition 48 .The theorem of Pythagoras - the converse. If in a triangle the square of one side equals the sum of the squares of the remaining sides, then the angle contained by the remaining sides is right.