Pappus’s Theorem
1
2
8
9
4
3
7
5
6
The collinearity of
(123), (456), (159), (168), (249), (267), (348), (357)
imples the collineartity of (7,8,9).
Pappus’s Theorem
1
2
8
9
4
3
7
5
6
The collinearity of
(123), (456), (159), (168), (249), (267), (348), (357)
imples the collineartity of (7,8,9).
Pappus’s Theorem
1
2
8
9
4
5
3
7
6
1
2
3
4
5
6
7
8
9
1
a
d
O
g
j
O
m
p
O
b
e
1
h
k
O
n
q
O
c
f
O
i
l
1
o
r
Another proof by algebraic cancellation
Pappus’s Theorem
1
2
8
9
4
5
3
7
6
1
2
3
4
5
6
7
8
9
1
a
d
O
g
j
O
m
p
O
b
e
1
h
k
O
n
q
O
c
f
O
i
l
1
o
r
(123)
(159)
(168)
(249)
(267)
(348)
(357)
(456)
(789)
Another proof by algebraic cancellation
==>
==>
==>
==>
==>
==>
==>
==>
==>
ce=bf
iq=hr
ko=ln
ar=cp
bj=ak
fm=do
dh=eg
gl=ij
mq=np
Pappus’s Theorem
1
2
8
9
4
5
3
7
6
1
2
3
4
5
6
7
8
9
1
a
d
O
g
j
O
m
p
O
b
e
1
h
k
O
n
q
O
c
f
O
i
l
1
o
r
(123)
(159)
(168)
(249)
(267)
(348)
(357)
(456)
(789)
Another proof by algebraic cancellation
==>
==>
==>
==>
==>
==>
==>
==>
==>
ce=bf
iq=hr
ko=ln
ar=cp
bj=ak
fm=do
dh=eg
gl=ij
mq=np
Pappus’s Theorem
1
2
8
9
4
5
3
7
6
1
2
3
4
5
6
7
8
9
1
a
d
O
g
j
O
m
p
O
b
e
1
h
k
O
n
q
O
c
f
O
i
l
1
o
r
(123)
(159)
(168)
(249)
(267)
(348)
(357)
(456)
(789)
Another proof by algebraic cancellation
==>
==>
==>
==>
==>
==>
==>
==>
<==
ce=bf
iq=hr
ko=ln
ar=cp
bj=ak
fm=do
dh=eg
gl=ij
mq=np
Generalizing this proof
(123)
(159)
(168)
(249)
(267)
(348)
(357)
(456)
(789)
==>
==>
==>
==>
==>
==>
==>
==>
<==
ce=bf
iq=hr
ko=ln
ar=cp
bj=ak
fm=do
dh=eg
gl=ij
mq=np
Problems:
special choice of basis
2x2 determinants
cancellation pattern
For Desargues’s Theorem there
is no such choice of a basis
Grassmann-Plücker
Relations
3
1
2
In every configuration of five points 1,2,3,x,y
[123][1xy]-[12x][13y]+[12y][13x] = O
is satisfied ( with [abc]=det(a,b,c) ).
Grassmann-Plücker
Relations
x
(123) collinear ==>
3
[12x][13y]=[12y][13x]
1
[12x][13y]=[12y][13x] ==>
(123) collinear or
2
(1xy) collinear
y
In every configuration of five points 1,2,3,x,y
[123][1xy]-[12x][13y]+[12y][13x] = O
is satisfied ( with [abc]=det(a,b,c) ).
Pappos’s Theorem
1
2
8
9
4
5
3
7
6
(123)
(159)
(168)
(249)
(456)
(348)
(267)
(357)
(789)
==>
==>
==>
==>
==>
==>
==>
==>
<==
[124][137]=[127][134]
[154][197]=[157][194]
[184][167]=[187][164]
[427][491]=[421][497]
[457][461]=[451][467]
[487][431]=[481][437]
[721][764]=[724][761]
[751][734]=[754][731]
[781][794]=[784][791]
Same cancellation pattern as before
Pappos’s Theorem
1
2
8
9
4
5
3
7
6
(123)
(159)
(168)
(249)
(456)
(348)
(267)
(357)
(789)
==>
==>
==>
==>
==>
==>
==>
==>
<==
[124][137]=[127][134]
[154][197]=[157][194]
[184][167]=[187][164]
[427][491]=[421][497]
[457][461]=[451][467]
[487][431]=[481][437]
[721][764]=[724][761]
[751][734]=[754][731]
[781][794]=[784][791]
Same cancellation pattern as before
Pappos’s Theorem
1
2
8
9
4
5
3
7
6
(123)
(159)
(168)
(249)
(456)
(348)
(267)
(357)
(789)
==>
==>
==>
==>
==>
==>
==>
==>
<==
[124][137]=[127][134]
[154][197]=[157][194]
[184][167]=[187][164]
[427][491]=[421][497]
[457][461]=[451][467]
[487][431]=[481][437]
[721][764]=[724][761]
[751][734]=[754][731]
[781][794]=[784][791]
Same cancellation pattern as before
Six points on a conic
2
1
3
6
4
5
Six points 1,2,3,4,5,6 are on a conic <==>
[123][156][426][453] = [456][126][254][423]
Pascal’s Theorem
2
1
3
8
9
6
7
4
5
[159][257]
[126][368]
[245][279]
[249][268]
[346][358]
[135][589]
[125][136][246][345]
=
=
=
=
=
=
=
-[125][579]
+[136][268]
-[249][257]
-[246][289]
+[345][368]
-[159][358]
+[126][135][245][346]
[289][579] = +[279][589]