Book I

Propositions

A proposition is either a problem (to construct) or a theorem (to prove). Walk each one step by step. The figure is drawn as Euclid would have drawn it — compass, then straightedge, then the reason.

• Proposition I.1

  The equilateral triangle

  On a given finite straight line, to construct an equilateral triangle.

  Walk the proof

• Proposition I.5

  Pons asinorum

  In isosceles triangles the angles at the base are equal to one another.

  Walk the proof

• Proposition I.15

  Vertical angles

  If two straight lines cut one another, they make the vertical angles equal.

  Walk the proof

• Proposition I.32

  The angle sum

  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 are equal to two right angles.

  Walk the proof

• Proposition I.47

  The Pythagorean theorem

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

  Walk the proof