Proposition 34

Theorem

The opposite sides and the opposite angles of a parallelogram are equal to one another, and either diagonal bisects the parallelogram.

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Commentary

Let ◇ABCD be a given parallelogram.

Then, the opposite sides are equal to each other, namely, ￜABￜ = ￜCDￜ and ￜBCￜ = ￜADￜ. The opposite angles are also equal to each other, namely, ∠CBA = ∠CDA and ∠DAB = ∠DCB.

Either of the two diagonals (AC or BD) bisects the parallelogram in area.

Euclid used the term "parallelogrammic area" here as a synonym for "parallelogram."

This proposition and the preceding one, Book 1 Proposition 33, give the principal properties of parallelograms.

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Computable version

In[1]:=

GeometricScene[{{A.,B.,C.,O.,D.,E.,F.,G.},{}},​​{GeometricStep[{CircleThrough[{A.,B.,C.},O.],Line[{A.,O.,C.}],Line[{D.,E.}],EuclideanDistance[D.,E.]≤EuclideanDistance[A.,C.]}],​​GeometricStep[{F.∈Line[{A.,C.}],EuclideanDistance[A.,F.]EuclideanDistance[D.,E.]}],​​GeometricStep[{GeometricAssertion[{CircleThrough[{F.},A.],CircleThrough[{A.,B.,C.},O.]},{"Concurrent",G.}]}],​​GeometricStep[{Line[{A.,G.}]}]},​​{EuclideanDistance[A.,G.]EuclideanDistance[D.,E.]}​​]//RandomInstance

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Additional instances

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