Euclid's Elements Book I

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

Proposition I.43 · Theorem

In any parallelogram the complements of the parallelograms about the diameter are equal to one another.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let ABCD be a parallelogram, and AC its diameter; about AC let EH, FG be parallelograms, and BK, KD the so-called complements. The claim: the complement BK is equal to the complement KD. (Try dragging A, B, D or K.)
  2. Since ABCD is a parallelogram, and AC its diameter, the triangle ABC is equal to the triangle ACD.
    Proposition I.34
    Proposition I.34: In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.
  3. Again, since EH is a parallelogram, and AK is its diameter, the triangle AEK is equal to the triangle AHK.
    Proposition I.34
    Proposition I.34: In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.
  4. For the same reason the triangle KFC is also equal to the triangle KGC.
    Proposition I.34
    Proposition I.34: In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.
  5. From the equal triangles ABC, ACD let the equal parts AEK, KGC and AHK, KFC be subtracted; then the complement BK which remains is equal to the complement KD which remains.
    Common Notion 3
    Common Notion 3: If equals be subtracted from equals, the remainders are equal.