Complete quadrilateral

A complete quadrilateral is a figure of Géométrie plane consisted of four right including two unspecified is not parallel nor three unspecified convergent.

Another manner of defining a complete quadrilateral is to supplement a convex Quadrilatère ABCD by the point E intersection of the right-hand sides (AB) and (CD) and the point F intersection of the right-hand sides (AD) and (BC) .

This figure is very related to the projective Géométrie and was studied as of IIe century by Menelaüs then Pappus of Alexandria.

Properties

Properties of the diagonals :
  • Each diagonal cuts the two others by creating harmonic divisions. In a more explicit way the diagonal (AC) cut the diagonal (dB) and the diagonal (EFF) in M and NR such as
\ frac {\ frac {\ overline {MY}} {\ overline {MC}}} {\ frac {\ overline {NA}} {\ overline {NC}}} =-1 It is a consequence of the Théorème of Ménélaüs and Théorème of Ceva
  • the mediums of the three diagonals are aligned on a line called Droite of Newton.

Theorem of Miquel : the circles circumscribed with the triangles (EAD) , (EBC) , (FAB) and (FDC) are convergent.

The Dual of the complete quadrilateral is the complete quadrangle

Remarkable use

The complete quadrangle registered in conical is very useful to show certain properties of the tangents and the Polaire S in a Conique.

See too

Related article

External bonds

  • complete quadrilateral on Chronomath
  • complete quadrilateral on the site of Debart

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