TXM800
LetΔbeabasicrationaltriangle.ThenXisarationalpointofΔiff
X
isalsoarationalpointofΔ.Inotherwords:​​(XΔ)⟺(
X
Δ)
NOTE:IntheRP-Theory​​
X
iscalledtheisogonalconjugateofthepointX.​​Atrianglewith3rationalsidesiscalledabasicrationaltriangle
EXAMPLE
In[]:=
x
1
=
2
(1+x)
(1+
2
x
) y (1+y)
2
(-1+x+y+x y)
2
(
2
y
+x y (1+y)+
2
x
(1+y+
2
y
))
(1+
2
x
) (
2
y
+
2
x
2
(1+y)
+x y (1+2 y));​​
y
1
=0;​​
x
2
=-
x
2
(1+x)
(-1+x (-1+y)-y) (1+y)
3
(-1+x+y+x y)
2
(
2
y
+x y (1+y)+
2
x
(1+y+
2
y
))
(1+
2
x
) (
2
y
+
2
x
2
(1+y)
+x y (1+2 y))
1+
2
x
;​​
y
2
=
2 x
2
(1+x)
(1+y) (x+y) (-1+x y)
2
(-1+x+y+x y)
2
(
2
y
+x y (1+y)+
2
x
(1+y+
2
y
))
(1+
2
x
) (
2
y
+
2
x
2
(1+y)
+x y (1+2 y))
1+
2
x
;​​
x
3
=0;​​
y
3
=0;​​
x
4
=
3
(1+x)
(1+y)
2
(x+y)
2
(-1+x+y+x y)
(y+x (-1+x+y+2 x y))
2
(
2
y
+x y (1+y)+
2
x
(1+y+
2
y
))
(1+
2
x
) (
2
y
+
2
x
2
(1+y)
+x y (1+2 y))
(1+
2
x
) (
2
y
+
2
x
2
(1+y)
+x y (1+2 y))
;​​
y
4
=2 x
2
(1+x)
(1+y) (x+y) (-1+x y)
2
(-1+x+y+x y)
(y+x (x+y+x y))
2
(
2
y
+x y (1+y)+
2
x
(1+y+
2
y
))
;​​
x
5
=(1+x) (1+
2
x
) (1+y) (x+y) (-1+x+y+x y) (x+y+x y) (
2
y
+x y (1+y)+
2
x
(1+y+
2
y
)) ((-1+y) y+x (1+y+2
2
y
)) (1+
2
x
) (
2
y
+
2
x
2
(1+y)
+x y (1+2 y));​​
y
5
=2 (1+x) (1+
2
x
) y (-1+x y) (-1+x+y+x y) (x+y+x y) (
2
y
+x (1+y+
2
y
)) (
2
y
+x y (1+y)+
2
x
(1+y+
2
y
)) (1+
2
x
) (
2
y
+
2
x
2
(1+y)
+x y (1+2 y));​​points=Table[{
x
n
,
y
n
},{n,1,5}];
NOTE:{
x
4
,
y
4
}and{
x
5
,
y
5
}aretwoisogonalconjugatepoints
In[]:=
Manipulate[Module[{pp,edges},pp=Chop[points/.{x->a,y->b}];edges=Subsets[pp,{2}];​​Graphics[{Tooltip[Line[#],EuclideanDistance@@#]&/@edges,Tooltip[{Red,Point[#]},#]&/@pp}]],​​{{a,1},-10,10,1/6,Appearance->"Labeled"},​​{{b,2},-10,10,1/7,Appearance->"Labeled"},SaveDefinitionsTrue]
Out[]=
​
a
1
b
2
Author: Minh Trinh Xuan