College Geometry: Construction of a parallelogram

THE PARALLELOGRAM

I°) Definition

I.1°) Definition

A parallelogram is a quadrilateral whose opposite sides are parallel.

I.2°) Construction

Let 3 points A, B and D. Draw the segments [AB] and [AD]. It then traces the line parallel to (AB) passing through point D and parallel to the right (AD) through the point B.

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We note C the intersection of these two lines. The quadrilateral ABCD is a parallelogram.

"ABCD is a parallelogram means that (AB)//(CD) and (AD)//(BC) .

II°) 1st characteristic property of the parallelogram.

The diagonals of a parallelogram intersect in the middle.

Conversely: :
If a quadrilateral has diagonals that intersect in the middle, then it is a parallelogram.

Construct a parallelogram with center O:

Let 3 points A, B and O. It traces the symmetrical around O of A and B, respectively named C and D.

O is the midpoint of [AC] and [BD], then ABCD is a parallelogram.

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III°) 2nd characteristic property of the parallelogram.

The opposite sides of a parallelogram have the same length.

Conversely:
if a quadrilateral has opposite sides of the same length, then it is a parallelogram.

Construction :

Draw [AB] and [AD], then we draw an arc of center D and radius AB, then an arc of center B and radius AD.

These two arcs intersect at C.

The opposite sides are of equal length, so that ABCD is a parallelogram.

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IV°) 3rd characteristic property of the parallelogram.

A quadrilateral with opposite sides parallel and equal length is a parallelogram.

Construction :

We built [AB] and a point D.

Draw the line parallel to (AB) passing through D, then the circle with center D and radius AB.

It intersects the parallel C

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(AB)//(CD) and AB = CD then ABCD is a parallelogram.