Proposition 33

Theorem

The lines (AC, BD) which join the adjacent endpoints of two equal and parallel lines (AB, CD) are equal and parallel.

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Commentary

Let AB and CD be two lines that are equal and parallel, and in the same direction (meaning that an arrow from A to B would point in the same direction as an arrow from C to D).

Connect A to C, and B to D, so that AC and BD are constructed.

Then AC and BD are also equal and parallel.

Euclid never formally defined the word "parallelogram," but this proposition can be considered the definition and construction of a parallelogram as a quadrilateral having one pair of opposite sides that are equal and parallel. This is equivalent to the modern definition of a parallelogram as a quadrilateral having each pair of opposite sides being parallel.

This proposition and the following one, Book 1 Proposition 34, 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

Out[]=

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Additional instances

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