Figure CabriII vers. MacOS 1.1.8

Window center x: 14.06_ y: 2.8

1: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Val: 0 0

2: Axes, 1, CN:1, VN:3
dGr, W, t, DS:1 1, GT:0, I, nSt
Const: 1, Val: 1 0, 0 1

3: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Val: -6.3 5.13_

4: Line, 0, CN:1, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 3, Val: 1 0

5: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Val: -1.23_ -3.73_

6: Par, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, V, nSt
Const: 5 4

"A0", NP: -24, 121, NS: 19, 17
7: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, V, nSt
Const: 6, Val: -0.53_ -3.73_
p: 0, Times, S: 12 C: 6 Fa: 0, p: 1, Times, S: 10 C: 6 Fa: 512

8: Cir, 0, CN:2, VN:2
Bl, W, t, DS:1 1, GT:0, I, nSt
Const: 5 7

9: Perp, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, V, nSt
Const: 5 6

10: Int, 256, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 9 8

11: Sym, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 5 7

12: Vec, 0, CN:2, VN:2
V, W, t, DS:1 1, GT:0, I, nSt
Const: 5 10

"A1", NP: -13, 76, NS: 19, 17
13: Tran, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 11 12
p: 0, Times, S: 12 C: 6 Fa: 0, p: 1, Times, S: 10 C: 6 Fa: 512

14: Refl, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 13 6

15: Refl, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 13 9

16: Refl, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 14 9

17: Con, 1, CN:5, VN:6
P, W, t, DS:1 1, GT:0, V, nSt
Const: 13 7 14 16 15

18: Sym, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 7 11

19: Vec, 0, CN:2, VN:2
V, W, t, DS:1 1, GT:0, I, nSt
Const: 5 11

20: Tran, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 18 19

21: Tran, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 20 19

22: Sym, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 5 10

23: Sym, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 5 22

24: Vec, 0, CN:2, VN:2
V, W, t, DS:1 1, GT:0, I, nSt
Const: 5 23

"A2", NP: 89, 14, NS: 19, 17
25: Tran, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 21 24
p: 0, Times, S: 12 C: 6 Fa: 0, p: 1, Times, S: 10 C: 6 Fa: 512

26: Line, 0, CN:2, VN:2
lGr, W, t, DS:1 1, GT:0, V, nSt
Const: 13 25

27: Par, 0, CN:2, VN:2
O, W, t, DS:1 1, GT:0, V, nSt
Const: 7 26

"A3", NP: 490, -217, NS: 19, 17
28: Int, 0, CN:3, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 27 17 7
p: 0, Times, S: 12 C: 6 Fa: 0, p: 1, Times, S: 10 C: 6 Fa: 512

29: Axes, 0, CN:3, VN:3
dGr, W, t, DS:1 1, GT:0, V, nSt
Const: 5 7 10

30: Line, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 13 28

31: Par, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 7 30

"A4", NP: 2000, -1064, NS: 19, 17
32: Int, 0, CN:3, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 31 17 7
p: 0, Times, S: 12 C: 6 Fa: 0, p: 1, Times, S: 10 C: 6 Fa: 512

33: Eq/Co, 1, CN:2, VN:3
B, W, NbD:2, FD, WU, GT:0, V, nSt
Const: 32 29, Val: 2051 -1046 56, nA, P
p: 0, Geneva, S: 9 C: 6 Fa: 0

34: Eq/Co, 0, CN:2, VN:3
B, W, NbD:2, FD, WU, GT:0, V, nSt
Const: 32 29, Val: 2011 -1046 97, nA, P
p: 0, Geneva, S: 9 C: 6 Fa: 0

35: Text, 2, CN:3, VN:3
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Const: 32 34 33, Val: 2006 -1046 0, A, nP, TP: 66.86_, 34.86_, TS: 2.76_, -0.4
"("#, "#)"
p: 0, Geneva, S: 9 C: 6 Fa: 0

36: Line, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 13 32

37: Par, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 7 36

"A5", NP: 7564, -4276, NS: 19, 17
38: Int, 0, CN:3, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 37 17 7
p: 0, Times, S: 12 C: 6 Fa: 0, p: 1, Times, S: 10 C: 6 Fa: 512

39: Eq/Co, 1, CN:2, VN:3
B, W, NbD:2, FD, WU, GT:0, V, nSt
Const: 38 29, Val: 7613 -4261 209, nA, P
p: 0, Geneva, S: 9 C: 6 Fa: 0

40: Eq/Co, 0, CN:2, VN:3
B, W, NbD:2, FD, WU, GT:0, V, nSt
Const: 38 29, Val: 7567 -4261 362, nA, P
p: 0, Geneva, S: 9 C: 6 Fa: 0

41: Text, 2, CN:3, VN:3
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Const: 38 40 39, Val: 7562 -4261 0, A, nP, TP: 252.06_, 142.03_, TS: 3.16_, -0.4
"("#, "#)"
p: 0, Geneva, S: 9 C: 6 Fa: 0

42: Eq/Co, 1, CN:2, VN:3
B, W, NbD:2, FD, WU, GT:0, V, nSt
Const: 28 29, Val: 555 -202 15, nA, P
p: 0, Geneva, S: 9 C: 6 Fa: 0

43: Eq/Co, 0, CN:2, VN:3
B, W, NbD:2, FD, WU, GT:0, V, nSt
Const: 28 29, Val: 515 -202 26, nA, P
p: 0, Geneva, S: 9 C: 6 Fa: 0

44: Text, 2, CN:3, VN:3
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Const: 28 43 42, Val: 510 -202 0, A, nP, TP: 17, 6.73_, TS: 2.76_, -0.4
"("#, "#)"
p: 0, Geneva, S: 9 C: 6 Fa: 0

45: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: -16 -142 0, A, nP, TP: -0.53_, 4.73_, TS: 7.53_, -3.13_
"Utilisation : rapprocher A0 de l'origine pour voir les points A4 et A5.
En formation, cette figure peut illustrer - de mani�re assez inattendue - la grande prŽcision de Cabri qui trouve ainsi, gŽomŽtriquement les premi�res solutions (enti�res) d'une Žquation de Pell Fermat."
p: 0, Times, S: 12 C: 9 Fa: 5, p: 11, Times, S: 12 C: 9 Fa: 0, p: 26, Times, S: 10 C: 9 Fa: 512, p: 27, Times, S: 12 C: 9 Fa: 0, p: 63, Times, S: 10 C: 9 Fa: 512, p: 64, Times, S: 12 C: 9 Fa: 0, p: 69, Times, S: 10 C: 9 Fa: 512, p: 70, Times, S: 12 C: 9 Fa: 0

46: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: 116 -374 0, A, nP, TP: 3.86_, 12.46_, TS: 11.33_, -0.63_
"Une utilisation de Cabri en arithmŽtique"
p: 0, Times, S: 18 C: 5 Fa: 1

47: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: -13 -298 0, A, nP, TP: -0.43_, 9.93_, TS: 11.6, -4.43_
"Dans un rep�re orthormormŽ du plan, on trace l'hyperbole ayant cette Žquation rŽduite, et on s'intŽresse aux solution enti�res positives x et y entiers naturels).On remarque que A0(1,0) et A1(2, 1) sont les premi�res solutions (c'est-ˆ-dire pour x=1 et x=2). La plus petite solution suivante - plus petite en x - est A2(7, 4).

Une partiedu sujet consiste ˆ montrer que si An (xn, yn) est une solution de l'Žquation , alors le point M  intersection de la parall�le ˆ (A1An) passant par A0  avec l'hyperbole est aussi une solution de l'Žquation. "
p: 0, Times, S: 12 C: 11 Fa: 0, p: 177, Times, S: 12 C: 6 Fa: 0, p: 179, Times, S: 10 C: 6 Fa: 512, p: 180, Times, S: 12 C: 6 Fa: 0, p: 185, Times, S: 12 C: 11 Fa: 0, p: 189, Times, S: 12 C: 6 Fa: 0, p: 190, Times, S: 10 C: 6 Fa: 512, p: 191, Times, S: 12 C: 6 Fa: 0, p: 197, Times, S: 12 C: 11 Fa: 0, p: 316, Times, S: 12 C: 6 Fa: 0, p: 318, Times, S: 10 C: 6 Fa: 512, p: 319, Times, S: 12 C: 6 Fa: 0, p: 325, Times, S: 12 C: 11 Fa: 0, p: 327, Times, S: 12 C: 6 Fa: 0, p: 374, Times, S: 10 C: 6 Fa: 512, p: 375, Times, S: 12 C: 6 Fa: 0, p: 378, Times, S: 10 C: 6 Fa: 512, p: 379, Times, S: 12 C: 6 Fa: 0, p: 382, Times, S: 10 C: 6 Fa: 512, p: 383, Times, S: 12 C: 6 Fa: 0, p: 433, Times, S: 12 C: 3 Fa: 0, p: 434, Times, S: 12 C: 6 Fa: 0, p: 435, Times, S: 12 C: 9 Fa: 0, p: 452, Times, S: 12 C: 2 Fa: 0, p: 464, Times, S: 12 C: 9 Fa: 0, p: 466, Times, S: 12 C: 13 Fa: 0, p: 469, Times, S: 10 C: 13 Fa: 512, p: 470, Times, S: 12 C: 13 Fa: 0, p: 471, Times, S: 10 C: 13 Fa: 512, p: 472, Times, S: 12 C: 13 Fa: 0, p: 473, Times, S: 12 C: 9 Fa: 0, p: 482, Times, S: 12 C: 2 Fa: 0, p: 487, Times, S: 10 C: 2 Fa: 512, p: 488, Times, S: 12 C: 9 Fa: 0, p: 494, Times, S: 12 C: 4 Fa: 0, p: 506, Times, S: 12 C: 9 Fa: 0, p: 517, Times, S: 12 C: 3 Fa: 0, p: 529, Times, S: 12 C: 9 Fa: 0, p: 530, Times, S: 12 C: 3 Fa: 0, p: 543, Times, S: 12 C: 6 Fa: 0

48: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: 59 -347 0, A, nP, TP: 1.96_, 11.56_, TS: 14.33_, -0.63_
"RŽsolution graphique l'Žquation de Pell-Fermat  x2 - 3y2 = 1 dans Z´Z."
p: 0, Times, S: 14 C: 3 Fa: 0, p: 47, Times, S: 14 C: 3 Fa: 1, p: 49, Times, S: 12 C: 3 Fa: 257, p: 50, Times, S: 14 C: 3 Fa: 1, p: 55, Times, S: 12 C: 3 Fa: 257, p: 56, Times, S: 14 C: 3 Fa: 1, p: 60, Times, S: 14 C: 3 Fa: 0, p: 67, Symbol, S: 14 C: 3 Fa: 0, p: 68, Times, S: 14 C: 3 Fa: 0

49: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: 295 -46 0, A, nP, TP: 9.83_, 1.53_, TS: 11.56_, -3.93_
"De plus on voit qu'avec la loi *, on a M = A1* An. 
Une autre question consiste alors ˆ montrer que ce point M est nŽcessairement An+1. (en un sens prŽcis mais qui revient ˆ dire que les solutions sont successives selon la croissance de x).

Si on prend toutes les solutions dans Z, qui s'obtiennent sur l'hyperbole en prenant les symŽtriques des points An par rapport aux axes de la conique, une autre question du sujet  montre que l'ensemble de ces solutions sur H est un sous-groupe de (H,* ).
"
p: 0, Times, S: 12 C: 11 Fa: 0, p: 39, Times, S: 12 C: 3 Fa: 0, p: 40, Times, S: 12 C: 11 Fa: 0, p: 43, Times, S: 12 C: 6 Fa: 0, p: 44, Times, S: 10 C: 6 Fa: 512, p: 45, Times, S: 12 C: 6 Fa: 0, p: 48, Times, S: 10 C: 6 Fa: 512, p: 49, Times, S: 12 C: 11 Fa: 0, p: 109, Times, S: 12 C: 3 Fa: 0, p: 110, Times, S: 12 C: 11 Fa: 0, p: 130, Times, S: 12 C: 6 Fa: 0, p: 131, Times, S: 10 C: 6 Fa: 512, p: 134, Times, S: 12 C: 11 Fa: 0, p: 241, Times, S: 12 C: 10 Fa: 0