Pappus's hexagon theorem

In mathematics, Pappus's hexagon theorem (attributed to <a href="/facts/Pappus_of_Alexandria/UABLMu0m">Pappus of Alexandria</a>) states that 

<ul><li>given one set of <a href="/facts/Collinearity/SxbVBJYR">collinear</a> points 
 
 
 
 A
 ,
 B
 ,
 C
 ,
 
 
 {\displaystyle A,B,C,}
 
 and another set of collinear points 
 
 
 
 a
 ,
 b
 ,
 c
 ,
 
 
 {\displaystyle a,b,c,}
 
 then the intersection points 
 
 
 
 X
 ,
 Y
 ,
 Z
 
 
 {\displaystyle X,Y,Z}
 
 of <a href="/facts/Line_(mathematics)/gJqsr4Gj">line</a> pairs 
 
 
 
 A
 b
 
 
 {\displaystyle Ab}
 
 and 
 
 
 
 a
 B
 ,
 A
 c
 
 
 {\displaystyle aB,Ac}
 
 and 
 
 
 
 a
 C
 ,
 B
 c
 
 
 {\displaystyle aC,Bc}
 
 and 
 
 
 
 b
 C
 
 
 {\displaystyle bC}
 
 are <a href="/facts/Collinear/SxbVBJYR">collinear</a>, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon 
 
 
 
 A
 b
 C
 a
 B
 c
 
 
 {\displaystyle AbCaBc}
 
.</li></ul>
It holds in a <a href="/facts/Projective_plane/p6ot8oNv">projective plane</a> over any field, but fails for projective planes over any noncommutative <a href="/facts/Division_ring/bCf07uRj">division ring</a>. Projective planes in which the "theorem" is valid are called pappian planes.
If one considers a pappian plane containing a hexagon as just described but with sides 
 
 
 
 A
 b
 
 
 {\displaystyle Ab}
 
 and 
 
 
 
 a
 B
 
 
 {\displaystyle aB}
 
 <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel</a> and also sides 
 
 
 
 B
 c
 
 
 {\displaystyle Bc}
 
 and 
 
 
 
 b
 C
 
 
 {\displaystyle bC}
 
 parallel (so that the Pappus line 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is the <a href="/facts/Line_at_infinity/RFnjNo29">line at infinity</a>), one gets the <a href="/facts/Affine_geometry/yUGgTx2n">affine version of Pappus's theorem</a> shown in the second diagram.
If the Pappus line 
 
 
 
 u
 
 
 {\displaystyle u}
 
 and the lines 
 
 
 
 g
 ,
 h
 
 
 {\displaystyle g,h}
 
 have a point in common, one gets the so-called little version of Pappus's theorem.
The <a href="/facts/Duality_(projective_geometry)/dxrjhRbB">dual</a> of this <a href="/facts/Intersection_theorem/UBgKPgtK">incidence theorem</a> states that given one set of <a href="/facts/Concurrent_lines/zPT6S4w9">concurrent lines</a> 
 
 
 
 A
 ,
 B
 ,
 C
 
 
 {\displaystyle A,B,C}
 
, and another set of concurrent lines 
 
 
 
 a
 ,
 b
 ,
 c
 
 
 {\displaystyle a,b,c}
 
, then the lines 
 
 
 
 x
 ,
 y
 ,
 z
 
 
 {\displaystyle x,y,z}
 
 defined by pairs of points resulting from pairs of intersections 
 
 
 
 A
 ∩
 b
 
 
 {\displaystyle A\cap b}
 
 and 
 
 
 
 a
 ∩
 B
 ,
 
 A
 ∩
 c
 
 
 {\displaystyle a\cap B,\;A\cap c}
 
 and 
 
 
 
 a
 ∩
 C
 ,
 
 B
 ∩
 c
 
 
 {\displaystyle a\cap C,\;B\cap c}
 
 and 
 
 
 
 b
 ∩
 C
 
 
 {\displaystyle b\cap C}
 
 are concurrent. (Concurrent means that the lines pass through one point.)
Pappus's theorem is a <a href="/facts/Special_case/bKaJ85XJ">special case</a> of <a href="/facts/Pascal%27s_theorem/61dfnDXG">Pascal's theorem</a> for a conic—the <a href="/facts/Limiting_case_(mathematics)/UEehGCom">limiting case</a> when the <a href="/facts/Degenerate_conic/9Ek4ja1g">conic degenerates</a> into 2 straight lines. Pascal's theorem is in turn a special case of the <a href="/facts/Cayley%E2%80%93Bacharach_theorem/HKbF7kvj">Cayley–Bacharach theorem</a>.
The <a href="/facts/Pappus_configuration/YYsP7euh">Pappus configuration</a> is the <a href="/facts/Projective_configuration/2EJN52Ud">configuration</a> of 9 lines and 9 points that occurs in Pappus's theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of intersection of 
 
 
 
 A
 B
 C
 
 
 {\displaystyle ABC}
 
 and 
 
 
 
 a
 b
 c
 
 
 {\displaystyle abc}
 
. This configuration is <a href="/facts/Duality_(projective_geometry)/dxrjhRbB">self dual</a>. Since, in particular, the lines 
 
 
 
 B
 c
 ,
 b
 C
 ,
 X
 Y
 
 
 {\displaystyle Bc,bC,XY}
 
 have the properties of the lines 
 
 
 
 x
 ,
 y
 ,
 z
 
 
 {\displaystyle x,y,z}
 
 of the dual theorem, and collinearity of 
 
 
 
 X
 ,
 Y
 ,
 Z
 
 
 {\displaystyle X,Y,Z}
 
 is equivalent to concurrence of 
 
 
 
 B
 c
 ,
 b
 C
 ,
 X
 Y
 
 
 {\displaystyle Bc,bC,XY}
 
, the dual theorem is therefore just the same as the theorem itself. The <a href="/facts/Levi_graph/pMnp7zkv">Levi graph</a> of the Pappus configuration is the <a href="/facts/Pappus_graph/w40dHpA6">Pappus graph</a>, a <a href="/facts/Bipartite_graph/xWcXV9MB">bipartite</a> <a href="/facts/Distance-regular_graph/pGAuLWYL">distance-regular</a> graph with 18 vertices and 27 edges.

Pappus's hexagon theorem open-in-new

Pappus's hexagon theorem