Marion's Theorem

Author

Eric W. Weisstein
November 23, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/MarionsTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MarionsTheorem.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Intersection Points

{aa,bb,cc}=IdentityMatrix[3]

bac={2c,0,a};
bca={c,0,2a};
acb={0,c,2b};
abc={0,2c,b};
cab={2b,a,0};
cba={b,2a,0};

d=Intersections[{bb,bac},{aa,acb}](-1)

e=Intersections[{cc,cab},{aa,abc}]

f=Intersections[{bb,bac},{cc,cba}]

g=Intersections[{aa,abc},{bb,bca}]

h=Intersections[{aa,acb},{cc,cba}](-1)

i=Intersections[{cc,cab},{bb,bca}](-1)

j=Intersections[{cc,cab},{bb,bac}]/(-2)

k=Intersections[{aa,abc},{bb,bac}]

l=Intersections[{aa,abc},{cc,cba}]/(-2)

m=Intersections[{bb,bca},{cc,cba}]

n=Intersections[{aa,acb},{bb,bca}]/2

o=Intersections[{cc,cab},{aa,acb}]

Figure

t=Triangle[{{0,0},{1,0},{.3,.6}}];

<<Graphics`Colors`

Show[Graphics[{
{Green,Polygon[TrilinearToCartesian[t,#]&/@#]&/@{{aa,d,j,e},{bb,f,l,g},{cc,h,n,i}}},
{Blue,Polygon[TrilinearToCartesian[t,#]&/@#]&/@{{aa,bac,d},{aa,cab,e},{bb,cba,f},{bb,abc,g},{cc,bca,i},{cc,acb,h}}},
{Purple,Polygon[TrilinearToCartesian[t,#]&/@#]&/@{{cab,e,k,f,cba},{abc,g,m,h,acb},{bca,i,o,d,bac}}},
{Red,Polygon[TrilinearToCartesian[t,#]&/@{j,k,l,m,n,o}]},
{Yellow,Polygon[TrilinearToCartesian[t,#]&/@#]&/@{{d,o,j},{i,o,n},{h,n,m},{f,k,l},{e,j,k},{g,l,m}}},
t,
Map[TrilinearToCartesian[t,#]&,Line[{First[#],KSection[Rest[#],3]}]&/@Permutations[IdentityMatrix[3]],{3}],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Transpose[{
Point/@(TrilinearToCartesian[t,#]&/@{abc,acb,bac,bca,cab,cba}),{"(Subscript[A, BC])","(Subscript[A, CB])","(Subscript[B, AC])","(Subscript[B, CA])","(Subscript[C, AB])","(Subscript[C, BA])"}}],
Transpose[{Point/@(TrilinearToCartesian[t,#]&/@{d,e,f,g,h,i,j,k,l,m,n,o}),
CharacterRange["D","O"]}]
],
LabelOffset->.025
]
}/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]
],AspectRatio->Automatic,PlotRange->All]

Areas

Green

FullSimplify[Plus@@Simplify[Abs/@Area[{aa,j,#}]&/@{d,e}]/Area[],TriangleQ[]]

FullSimplify[Plus@@Simplify[Abs/@Area[{bb,l,#}]&/@{f,g}]/Area[],TriangleQ[]]

FullSimplify[Plus@@Simplify[Abs/@Area[{cc,n,#}]&/@{h,i}]/Area[],TriangleQ[]]

Blue

Abs/@(Area/@{{aa,bac,d},{aa,cab,e},{bb,cba,f},{bb,abc,g},{cc,bca,i},{cc,acb,h}}/Area[])

Purple

Plus@@Abs/@Simplify[Area/@{{cab,e,k},{cba,f,k},{cab,cba,k}}/Area[]]

Plus@@Abs/@Simplify[Area/@{{abc,g,m},{acb,h,m},{abc,acb,m}}/Area[]]

Plus@@Abs/@Simplify[Area/@{{bac,d,o},{bca,i,o},{bca,bac,o}}/Area[]]

Yellow

Abs/@Simplify[Area/@{{d,o,j},{i,o,n},{h,n,m},{f,k,l},{e,j,k},{g,l,m}}/Area[]]

Red

1-(3/14+3 11/105+6/21+6/70)

Morgan's Theorem

Table[(8/((3n+1)(3n-1))),{n,20}]

Through[{Numerator,Denominator}[%]]

Table[(8/((3n+1)(3n-1))),{n,1,20,2}]

Denominator[%]

Table[(3n-1)(3n-2)/2,{n,10}]