Euclid's Elements Book I

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

Proposition I.7 · Theorem

Given two straight lines constructed on a straight line from its extremities and meeting in a point, there cannot be constructed on the same straight line from its extremities, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. On the straight line AB let two straight lines AC, CB be constructed from its ends A, B, meeting at the point C. The claim: on the same side of AB there cannot be constructed from A, B two other straight lines AD, DB meeting at another point D, with AD equal to AC and DB equal to CB.
  2. Suppose it possible: let AD be equal to AC and DB to CB, with D a different point on the same side. (Such a D cannot really be drawn; the D shown is only a sketch, and you can drag it.)
  3. Let CD be joined. Since AC is equal to AD, the angle ACD is equal to the angle ADC.
    Postulate 1Proposition I.5
    Postulate 1: To draw a straight line from any point to any point.
    Proposition I.5: 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.
  4. Therefore the angle ADC is greater than the angle DCB; therefore the angle CDB is much greater than the angle DCB.
    Common Notion 5
    Common Notion 5: The whole is greater than the part.
  5. Again, since CB is equal to DB, the angle CDB is equal to the angle DCB. But it was also proved much greater than it, which is impossible. So no such D exists.
    Proposition I.5
    Proposition I.5: 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.
  6. To see it directly: the circle with centre A and radius AC and the circle with centre B and radius BC meet on this side of AB only at C.