Figure CabriII vers. MacOS 1.1.5

Used macro(s):

Droite de Simson, no name
Icon:
0FFFFFFFFF000000
000FF0000FFF0000
000FF000000FF000
000FF0000000FF00
000FF0000000FF00
000FF00000000FF0
000FF00000000FF0
000FF00000000FF0
000FF00000000FF0
000FF00000000FF0
000FF00000000FF0
000FF0000000FF00
000FF0000000FF00
000FF000000FF000
000FF0000FFF0000
0FFFFFFFFF000000
Mth: 0
CN:4, ON:7, FN:2, PO:6
CT:
point, CS 0, R, W, t, DS:1 1, GT:1, V, nSt
point, CS 0, R, W, t, DS:1 1, GT:1, V, nSt
point, CS 0, R, W, t, DS:1 1, GT:1, V, nSt
point, CS 0, Bl, W, t, DS:1 1, GT:2, V, nSt
Const:
Line, Mth:1, 0, 0, CN:2, VN:2, Const: 2 3
Line, Mth:1, 0, 0, CN:2, VN:2, Const: 3 1
Perp, Mth:0, 0, 0, CN:2, VN:2, Const: 4 5
Perp, Mth:0, 0, 0, CN:2, VN:2, Const: 4 6
Int, Mth:0, 0, 0, CN:2, VN:1, Const: 7 5
Int, Mth:0, 1, 0, CN:2, VN:1, Const: 8 6, R, W, t, DS:1 1, GT:1, I, nSt
Line, Mth:1, 1, 0, CN:2, VN:2, Const: 9 10, P, W, t, DS:1 1, GT:0, V, nSt

Figure description:

Window center x: -1 y: 0.16_

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
Y, W, t, DS:1 1, GT:0, I, nSt
Const: 1, Val: 1 0, 0 1

"A", NP: -130, -122, NS: 14, 12
3: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Val: -4.53_ 3.53_
p: 0, Times, S: 12 C: 9 Fa: 0

"B", NP: -153, 46, NS: 14, 12
4: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Val: -5.3 -2.03_
p: 0, Times, S: 12 C: 9 Fa: 0

"C", NP: -33, 55, NS: 14, 12
5: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Val: -0.96_ -1.53_
p: 0, Times, S: 12 C: 9 Fa: 0

6: Tr, 0, CN:3, VN:4
dG, W, t, DS:1 1, GT:0, V, nSt
Const: 3 4 5

7: PBiss, 2, CN:1, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 6

8: PBiss, 0, CN:1, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 6

"O", NP: -104, -12, NS: 15, 12
9: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 7 8
p: 0, Times, S: 12 C: 6 Fa: 1

10: Cir, 0, CN:2, VN:2
Bl, W, t, DS:1 1, GT:0, V, nSt
Const: 9 3

"M", NP: -12, -69, NS: 17, 12
11: Pt/, 0, CN:1, VN:3
Bl, W, t, DS:1 1, GT:2, V, nSt
Const: 10, Val: -0.628190879949148 2.13549992957587
p: 0, Times, S: 12 C: 6 Fa: 1

12: Line, 0, CN:2, VN:2
dG, W, t, DS:1 1, GT:0, V, nSt
Const: 4 5

13: Line, 0, CN:2, VN:2
dG, W, t, DS:1 1, GT:0, V, nSt
Const: 5 3

14: Line, 0, CN:2, VN:2
dG, W, t, DS:1 1, GT:0, V, nSt
Const: 3 4

15: Perp, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 11 12

16: Perp, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 11 13

17: Perp, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 11 14

"A'", NP: 0, 30, NS: 18, 12
18: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 15 12
p: 0, Times, S: 12 C: 4 Fa: 0

"B'", NP: -72, -31, NS: 18, 12
19: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 16 13
p: 0, Times, S: 12 C: 4 Fa: 0

"C'", NP: -160, -85, NS: 18, 12
20: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 14 17
p: 0, Times, S: 12 C: 4 Fa: 0

"S(M)", NP: -180, -129, NS: 31, 12
21: Line, 0, CN:2, VN:2
P, W, t, DS:1 1, GT:0, V, nSt
Const: 18 19
p: 0, Times, S: 12 C: 4 Fa: 0

22: Sym, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 4 18

23: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: 40 -172 1, A, nP, TP: 1.33_, 5.73_, TS: 8.56_, -3
"Angle de deux droites de Simson

Soient ABC un triangle, M et N deux points de son cercle circonscrit, de centre O.

Alors 2(S(M), S(N)) = (ON, OM)"
p: 0, Times, S: 14 C: 5 Fa: 5, p: 31, Times, S: 14 C: 5 Fa: 1, p: 33, Times, S: 14 C: 5 Fa: 0, p: 116, Times, S: 14 C: 3 Fa: 0

"N", NP: -190, -47, NS: 16, 12
24: Pt/, 0, CN:1, VN:3
O, W, t, DS:1 1, GT:2, V, nSt
Const: 10, Val: -6.52216846881652 1.24474972748272
p: 0, Times, S: 12 C: 2 Fa: 1

Ma: Droite de Simson, Const: 3 i: 0 4 i: 0 5 i: 0 24 i: 0
30: Ma R, F No1, VN:1
R, W, t, DS:1 1, GT:1, I, nSt

"S(N)", NP: -100, -143, NS: 30, 12
31: Ma R, F No2, VN:2
O, W, t, DS:1 1, GT:0, V, nSt
p: 0, Times, S: 12 C: 2 Fa: 0

"C""", NP: -141, -37, NS: 19, 12
32: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 14 31
p: 0, Times, S: 12 C: 2 Fa: 0

"A""", NP: -181, 68, NS: 19, 12
33: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 31 12
p: 0, Times, S: 12 C: 2 Fa: 0

"B""", NP: -115, -92, NS: 19, 12
34: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 13 31
p: 0, Times, S: 12 C: 2 Fa: 0

35: Line, 0, CN:2, VN:2
lBl, W, t, DS:1 1, GT:0, V, nSt
Const: 11 18

36: Line, 0, CN:2, VN:2
lBl, W, t, DS:1 1, GT:0, V, nSt
Const: 24 33

"U", NP: -5, 5, NS: 15, 12
37: Int, 0, CN:3, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 35 10 11
p: 0, Times, S: 12 C: 7 Fa: 0

"V", NP: -186, 21, NS: 14, 12
38: Int, 0, CN:3, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 36 10 24
p: 0, Times, S: 12 C: 7 Fa: 0

39: Ray, 0, CN:2, VN:2
lGr, W, t, DS:1 1, GT:0, V, nSt
Const: 3 37

40: Ray, 0, CN:2, VN:2
lGr, W, t, DS:1 1, GT:0, V, nSt
Const: 3 38

41: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: 40 -61 1, A, nP, TP: 1.33_, 2.03_, TS: 7.1, -2
"On note U et V les intersections de (MA') et (NA"") avec le cercle circonscrit.

Pour la preuve, utiliser que
(S(M), S(N)) = (AU, AV)."
p: 0, Times, S: 12 C: 10 Fa: 0, p: 80, Times, S: 12 C: 9 Fa: 0

42: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 21 31

43: Angle, 0, CN:3, VN:5
P, W, t, DS:1 1, GT:24, V, nSt
Const: 33 42 18, Val: -1, 1

44: Angle, 0, CN:3, VN:5
Br, W, t, DS:1 1, GT:24, V, nSt
Const: 11 9 24, Val: 1, 1

45: AngVal, 0, CN:1, VN:1
P, W, NbD:-5, FD, ”, GT:0, V, nSt
Const: 43, Val: -99 -83 1.1991, nA, P
p: 0, Geneva, S: 9 C: 4 Fa: 0

46: Text, 0, CN:2, VN:3
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Const: 43 45, Val: -100 -83 0, nA, nP, TP: -3.33_, 2.76_, TS: 1.93_, -0.4
""#"
p: 0, Geneva, S: 9 C: 6 Fa: 0

47: AngVal, 0, CN:1, VN:1
Br, W, NbD:-5, FD, ”, GT:0, V, nSt
Const: 44, Val: -125 1 2.3982, nA, P
p: 0, Geneva, S: 9 C: 11 Fa: 0

48: Text, 0, CN:2, VN:3
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Const: 44 47, Val: -126 1 0, nA, nP, TP: -4.2, -0.03_, TS: 2.13_, -0.4
""#"
p: 0, Geneva, S: 9 C: 6 Fa: 0

49: Seg, 0, CN:2, VN:2
Br, W, t, DS:1 1, GT:0, V, nSt
Const: 9 11

50: Seg, 0, CN:2, VN:2
Br, W, t, DS:1 1, GT:0, V, nSt
Const: 9 24

51: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: 39 46 1, A, nP, TP: 1.3, -1.53_, TS: 8.3, -1.2
"En particulier deux droites de Simson S(M) et S(N) sont orthogonales ssi les points M et N sont diamŽtralement opposŽs sur le cercle circonscrit."
p: 0, Times, S: 12 C: 6 Fa: 0