Rhombus

In a Espace refines normalized, a rhombus is a Parallélogramme having two consecutive sides of the same length.

Properties

Property 1

For all Quadrilatère of a plan refines Euclidean (Espace refines Euclidean dimension 2) the following proposals are equivalent:

  1. the rhombus is a quadrilateral.

  2. the rhombus has its four of the same sides length.
  3. the rhombus is a parallelogram and its diagonals are perpendicular.

These equivalences are however at fault in the case of a rhombus flattened (item 3 then does not have a direction):

Proof

That is to say ABCD a quadrilateral. That is to say I medium of and J medium of.

As \ neq C has one can speak about the Médiatrice d_ {AC} about. As B \ neq D one can speak about the Médiatrice d_ {data base} about.

Let us show (1) implies (2):

It is supposed that ABCD is a rhombus.

As it is a parallelogram, one has AB = CD, BC = AD and as it is a rhombus, one has AB = CB. By transitivity, AB = BC = CD = DA.

Let us show (2) implies (3):

It is supposed that AB = BC = CD = DA.

AB = BC and CD = DA, one concludes (dB) = d_ {AC} . Thus (dB) is perpendicular to (AC) and I belongs to (dB) and (AC).

BC = CD, one concludes that C \ in d_ {data base} .

There is (dB) \ perp (AC) and (d_ {data base}) \ perp (data base) thus (d_ {data base}) \ parallel (AC) . As d_ {data base} and (AC) has the point C joint, it is concluded that d_ {data base} = (AC) and thus that J belongs to (AC) and (data base).

As (AC) and (data base) are perpendicular, they have a single common point and thus I = J. ABCD has its diagonals which are cut in their medium, it is thus a parallelogram.

Let us show (3) implies (1):

It is supposed that (AC) and (data base) are perpendicular and that ABCD is a parallelogram. As (AC) is perpendicular to (data base) and passes by J, it is concluded that (AC) = d_ {data base} and thus that CB = CD.

Property 2

The diagonals of a rhombus are the bisectrices of its angles.

Proof

That is to say a rhombus ABCD of center O. property 1 involves that triangles ABO, CBO, TEENAGER and CDO are superposable. From where:

\ widehat {OAB} = \ widehat {OAD} = \ widehat {OCB} = \ widehat {OCD}

\ widehat {OBA} = \ widehat {OBC} = \ widehat {ODA} = \ widehat {ODC}

i.e. the diagonals of the rhombus are the bisectrices of its angles.

Property 3

The opposite angles of a rhombus have same measurement two to two.

Proof

That is to say a rhombus ABCD of center O. According to the proof of property 2:

\ widehat {OAB} = \ widehat {OAD} = \ widehat {OCB} = \ widehat {OCD}

\ widehat {OBA} = \ widehat {OBC} = \ widehat {ODA} = \ widehat {ODC}

Therefore, \ widehat {DAB} = \ widehat {DCB} and \ widehat {ABC} = \ widehat {ADC} .

Property 4

A rhombus has at least two axes of symmetry: its diagonals.

Proof

That is to say a rhombus ABCD of center O. According to 3. property 1, the diagonals are cut in their medium (property of the Parallélogramme) and are perpendicular. Thus C is the image of has by the symmetry of axis (data base) and D is the image of B by the symmetry of axis (AC).

Notice

The definition of the rhombus as being a parallelogram imposes that a rhombus is a plane figure. There exist quadrilaterals (with four quite distinct tops) having the four of the same sides length which are not rhombuses. It is enough to be placed in a Espace refines Euclidean dimension 3 and to subject to a side of a " truth losange" a rotation according to one as of its diagonals.

Surface

If has and B is the lengths of the diagonals, then the surface of the rhombus is:
A= \ frac {has \ times B} {2}
indeed, the diagonals define four Triangle S rectangles which it is enough to réagencer to have a rectangle whose sides are has /2 and B (for example); one applies the formula then giving the surface of the rectangle.

Rhombohedron

A Rhomboèdre is a Polyèdre whose six faces are rhombuses.

Anecdote

“The Rhombus” or “the mark with the rhombus” is expressions regularly used to indicate the automobile mark Renault, by analogy with the form of its logo.

The rhombus, symbol of life and prosperity, is the base form of watch DELANCE, it even symbol of the creative power to the female one. (Gisele Rufer-Delance)

Simple: Rhombus

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