Euclid's Elements Book I

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

Proposition I.11 · Problem

To draw a straight line at right angles to a given straight line from a given point on it.

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 and C the given point on it (you can drag it). It is required to draw from the point C a straight line at right angles to AB.
  2. Let a point D be taken at random on AC (you can drag it), and let CE be made equal to CD.
    Proposition I.3
    Proposition I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
  3. On DE let the equilateral triangle FDE be constructed.
    Proposition I.1
    Proposition I.1: On a given finite straight line to construct an equilateral triangle.
  4. Let FC be joined. The claim: FC is the straight line drawn from C at right angles to AB.
    Postulate 1
    Postulate 1: To draw a straight line from any point to any point.
  5. Since DC is equal to CE, and CF is common, the two sides DC, CF are equal to the two sides EC, CF, and the base DF is equal to the base FE; therefore the angle DCF is equal to the angle ECF.
    Proposition I.8
    Proposition I.8: If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines.
  6. And they are adjacent. When a straight line set up on a straight line makes the adjacent angles equal, each of them is right; therefore each of the angles DCF, FCE is right, and FC is at right angles to AB.
    Definition 10
    Definition 10: When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.