Editing Quadrilateral (section)

==Remarkable points and lines in a convex quadrilateral==
The centre of a quadrilateral can be defined in several different ways. The "vertex centroid" comes from considering the quadrilateral as being empty but having equal masses at its vertices. The "side centroid" comes from considering the sides to have constant mass per unit length. The usual centre, called just [[centroid]] (centre of area) comes from considering the surface of the quadrilateral as having constant density. These three points are in general not all the same point.<ref>{{Cite web|url=https://sites.math.washington.edu/~king/java/gsp/center-mass-quad.html|title=Two Centers of Mass of a Quadrilateral|website=Sites.math.washington.edu|access-date=1 March 2022}}</ref>

The "vertex centroid" is the intersection of the two [[Quadrilateral#Special line segments|bimedians]].<ref>Honsberger, Ross, ''Episodes in Nineteenth and Twentieth Century Euclidean Geometry'', Math. Assoc. Amer., 1995, pp. 35–41.</ref> As with any polygon, the ''x'' and ''y'' coordinates of the vertex centroid are the [[arithmetic mean]]s of the ''x'' and ''y'' coordinates of the vertices.

The "area centroid" of quadrilateral ''ABCD'' can be constructed in the following way. Let ''G<sub>a</sub>'', ''G<sub>b</sub>'', ''G<sub>c</sub>'', ''G<sub>d</sub>'' be the centroids of triangles ''BCD'', ''ACD'', ''ABD'', ''ABC'' respectively. Then the "area centroid" is the intersection of the lines ''G<sub>a</sub>G<sub>c</sub>'' and ''G<sub>b</sub>G<sub>d</sub>''.<ref name=Myakishev>{{citation
 | last = Myakishev
 | first = Alexei
 | journal = Forum Geometricorum
 | pages = 289–295
 | title = On Two Remarkable Lines Related to a Quadrilateral
 | url = http://forumgeom.fau.edu/FG2006volume6/FG200634.pdf
 | volume = 6
 | year = 2006
 | access-date = 2012-04-15
 | archive-date = 2019-12-31
 | archive-url = https://web.archive.org/web/20191231055834/http://forumgeom.fau.edu/FG2006volume6/FG200634.pdf
 | url-status = dead
 }}.</ref>

In a general convex quadrilateral ''ABCD'', there are no natural analogies to the [[circumcenter]] and [[orthocenter]] of a [[triangle]]. But two such points can be constructed in the following way. Let ''O<sub>a</sub>'', ''O<sub>b</sub>'', ''O<sub>c</sub>'', ''O<sub>d</sub>'' be the circumcenters of triangles ''BCD'', ''ACD'', ''ABD'', ''ABC'' respectively; and denote by ''H<sub>a</sub>'', ''H<sub>b</sub>'', ''H<sub>c</sub>'', ''H<sub>d</sub>'' the orthocenters in the same triangles. Then the intersection of the lines ''O<sub>a</sub>O<sub>c</sub>'' and ''O<sub>b</sub>O<sub>d</sub>'' is called the [[circumcenter of mass|quasicircumcenter]], and the intersection of the lines ''H<sub>a</sub>H<sub>c</sub>'' and ''H<sub>b</sub>H<sub>d</sub>'' is called the ''quasiorthocenter'' of the convex quadrilateral.<ref name=Myakishev/> These points can be used to define an [[Euler line]] of a quadrilateral. In a convex quadrilateral, the quasiorthocenter ''H'', the "area centroid" ''G'', and the quasicircumcenter ''O'' are [[collinear]] in this order, and ''HG'' = 2''GO''.<ref name=Myakishev/>

There can also be defined a ''quasinine-point center'' ''E'' as the intersection of the lines ''E<sub>a</sub>E<sub>c</sub>'' and ''E<sub>b</sub>E<sub>d</sub>'', where ''E<sub>a</sub>'', ''E<sub>b</sub>'', ''E<sub>c</sub>'', ''E<sub>d</sub>'' are the [[Nine-point circle|nine-point centers]] of triangles ''BCD'', ''ACD'', ''ABD'', ''ABC'' respectively. Then ''E'' is the [[midpoint]] of ''OH''.<ref name=Myakishev/>

Another remarkable line in a convex non-parallelogram quadrilateral is the [[Newton line]], which connects the midpoints of the diagonals, the segment connecting these points being bisected by the vertex centroid. One more interesting line (in some sense dual to the [[Newton line|Newton's]] one) is the line connecting the point of intersection of diagonals with the vertex centroid. The line is remarkable by the fact that it contains the (area) centroid. The vertex centroid divides the segment connecting the intersection of diagonals and the (area) centroid in the ratio 3:1.<ref>{{cite web|url=https://www.austms.org.au/Publ/Gazette/2010/May10/TechPaperMiller.pdf|title=Centroid of a quadrilateral|author=John Boris Miller|website=Austmd.org.au|access-date=March 1, 2022}}</ref>

For any quadrilateral ''ABCD'' with points ''P'' and ''Q'' the intersections of ''AD'' and ''BC'' and ''AB'' and ''CD'', respectively, the circles ''(PAB), (PCD), (QAD),'' and ''(QBC)'' pass through a common point ''M'', called a Miquel point.<ref>{{Cite book|title=Euclidean Geometry in Mathematical Olympiads|last=Chen|first=Evan|author-link=Evan Chen|publisher=Mathematical Association of America|year=2016|isbn=9780883858394|location=Washington, D.C.|pages=198}}</ref>

For a convex quadrilateral ''ABCD'' in which ''E'' is the point of intersection of the diagonals and ''F'' is the point of intersection of the extensions of sides ''BC'' and ''AD'', let ω be a circle through ''E'' and ''F'' which meets ''CB'' internally at ''M'' and ''DA'' internally at ''N''. Let ''CA'' meet ω again at ''L'' and let ''DB'' meet ω again at ''K''. Then there holds: the straight lines ''NK'' and ''ML'' intersect at point ''P'' that is located on the side ''AB''; the straight lines ''NL'' and ''KM'' intersect at point ''Q'' that is located on the side ''CD''. 
Points ''P'' and ''Q'' are called "Pascal points" formed by circle ω on sides ''AB'' and ''CD''.
<ref name=Fraivert>{{citation
 | last = David
 | first = Fraivert
 | journal = [[The Mathematical Gazette]]
 | pages = 233–239
 | title = Pascal-points quadrilaterals inscribed in a cyclic quadrilateral
 | volume = 103
 | year = 2019| issue = 557
 | doi = 10.1017/mag.2019.54
 | s2cid = 233360695
 }}.</ref>
<ref name=Fraivert2>{{citation
 | last = David
 | first = Fraivert
 | journal = Journal for Geometry and Graphics
 | pages = 5–27
 | title = A Set of Rectangles Inscribed in an Orthodiagonal Quadrilateral and Defined by Pascal-Points Circles
 | url = http://www.heldermann.de/JGG/JGG23/JGG231/jgg23002.htm
 | volume = 23
 | year = 2019}}.</ref>
<ref name=Fraivert3>{{citation
 | last = David
 | first = Fraivert
 | journal = [[Forum Geometricorum]]
 | pages = 509–526
 | title = Properties of a Pascal points circle in a quadrilateral with perpendicular diagonals
 | url = http://forumgeom.fau.edu/FG2017volume17/FG201748.pdf
 | volume = 17
 | year = 2017
 | access-date = 2020-04-29
 | archive-date = 2020-12-05
 | archive-url = https://web.archive.org/web/20201205215507/http://forumgeom.fau.edu/FG2017volume17/FG201748.pdf
 | url-status = dead
 }}.</ref>