Euclid's Elements Book I

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

Proposition I.46 · Problem

On a given straight line to describe a square.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let AB be the given straight line. It is required to describe a square on AB. (Try dragging A or B.)
  2. From the point A on the straight line AB let AC be drawn at right angles to AB.
    Proposition I.11
    Proposition I.11: To draw a straight line at right angles to a given straight line from a given point on it.
  3. On it let AD be cut off equal to AB.
    Proposition I.3
    Proposition I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
  4. Through D let DE be drawn parallel to AB, and through B let BE be drawn parallel to AD.
    Proposition I.31
    Proposition I.31: Through a given point to draw a straight line parallel to a given straight line.
  5. Then ADEB is a parallelogram; therefore AB is equal to DE, and AD to BE. But AB is equal to AD; therefore the four sides BA, AD, DE, EB are equal to one another.
    Proposition I.34Common Notion 1
    Proposition I.34: In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.
    Common Notion 1: Things which are equal to the same thing are also equal to one another.
  6. Again, since AD falls on the parallels AB, DE, the angles BAD, ADE are equal to two right angles. But the angle BAD is right; therefore the angle ADE is also right, and the opposite angles of a parallelogram are equal, so all four angles are right. Therefore ADEB is the square described on AB.
    Proposition I.29Proposition I.34Definition 22
    Proposition I.29: A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.
    Proposition I.34: In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.
    Definition 22: Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.