Euclid's Elements Book I

23 definitions · 5 postulates · 5 common notions · 48 propositions

Proposition I.10 · Problem

To bisect a given finite straight line.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let AB be the given finite straight line. It is required to bisect the finite straight line AB.
  2. On AB let the equilateral triangle ABC be constructed.
    Proposition I.1
    Proposition I.1: On a given finite straight line to construct an equilateral triangle.
  3. Let the angle ACB be bisected by the straight line CD, meeting AB at D. The claim: the straight line AB has been bisected at the point D.
    Proposition I.9
    Proposition I.9: To bisect a given rectilineal angle.
  4. Since AC is equal to CB, and CD is common, the two sides AC, CD are equal to the two sides BC, CD respectively; and the angle ACD is equal to the angle BCD; therefore the base AD is equal to the base BD. So the given finite straight line AB has been bisected at D.
    Proposition I.4
    Proposition I.4: 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.