Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms

(Proposition 34 from Book 1 of Euclid's “Elements”)

In parallelogrammic figures the opposite sides and angles are equal to one another, and a diagonal cuts them in half. * Let $ACDB$ be a parallelogrammic figure, and $BC$ its diagonal. * I say that for parallelogram $ACDB$, the opposite sides and angles are equal to one another, and the diagonal $BC$ cuts it in half.

Modern Formulation

The opposite sides and the opposite angles of a parallelogram are equal to one another and either diagonal bisects the parallelogram. In particular, the area of the parallelogram $$\boxdot{ABDC}$$ is double the area of $$\triangle{ACB}$$, (respectively $$\triangle{BCD}$$).

Proofs: 1 Corollaries: 1 2 3 4 5 6

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References

Adapted from CC BY-SA 3.0 Sources:

1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014

Adapted from (Public Domain)

1. Casey, John: "The First Six Books of the Elements of Euclid"

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"