Euclid's Elements Book I

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

Proposition I.34 · Theorem

In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let ACDB be a parallelogrammic area, and BC its diameter. The claim: the opposite sides and angles of the parallelogram ACDB are equal to one another, and the diameter BC bisects it. (Try dragging A, B, C.)
  2. Let the diameter BC be joined.
    Postulate 1
    Postulate 1: To draw a straight line from any point to any point.
  3. Since AB is parallel to CD, and the straight line BC has fallen upon them, the alternate angles ABC, BCD are equal to one another.
    Proposition I.29
    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.
  4. Again, since AC is parallel to BD, and BC has fallen upon them, the alternate angles ACB, CBD are equal to one another.
    Proposition I.29
    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.
  5. So ABC, DCB are two triangles having the two angles ABC, BCA equal to the two angles DCB, CBD respectively, and the side BC between them common; therefore AB is equal to CD, AC to BD, and the angle BAC to the angle CDB.
    Proposition I.26
    Proposition I.26: If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle to the remaining angle.
  6. And since the angle ABC is equal to the angle BCD, and the angle CBD to the angle ACB, the whole angle ABD is equal to the whole angle ACD.
    Common Notion 2
    Common Notion 2: If equals be added to equals, the wholes are equal.
  7. Since AB is equal to CD and BC is common, and the angle ABC is equal to the angle BCD, the triangle ABC is equal to the triangle DCB; therefore the diameter BC bisects the parallelogram ACDB.
    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.