Figure CabriII vers. MacOS 1.1.5

Window center x: 1.8 y: 0.66_

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: -84, -104, NS: 14, 12
3: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Val: -2.9 2.96_
p: 0, Times, S: 12 C: 4 Fa: 0

"B", NP: -151, 62, NS: 14, 12
4: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Val: -4.7 -1.96_
p: 0, Times, S: 12 C: 4 Fa: 0

"C", NP: 45, 65, NS: 14, 12
5: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Val: 1.43_ -1.96_
p: 0, Times, S: 12 C: 4 Fa: 0

6: Tr, 0, CN:3, VN:4
P, 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: -44, 7, NS: 14, 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: 0

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

11: Seg, 0, CN:2, VN:2
Gr, W, t, DS:1 1, GT:0, V, nSt
Const: 9 3

"t", NP: -36, -124, NS: 13, 15
12: Perp, 0, CN:2, VN:2
Br, W, t, DS:1 1, GT:0, V, nSt
Const: 3 11
p: 0, Symbol, S: 14 C: 10 Fa: 0

"Æ", NP: -88, -132, NS: 16, 15
13: Biss, 0, CN:3, VN:2
G, W, t, DS:1 1, GT:0, V, nSt
Const: 4 3 5
p: 0, Times, S: 14 C: 9 Fa: 0

"d", NP: 83, 43, NS: 14, 15
14: Line, 0, CN:2, VN:2
P, W, t, DS:1 1, GT:0, V, nSt
Const: 4 5
p: 0, Times, S: 14 C: 4 Fa: 0

15: Refl, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 4 13

16: Refl, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 5 13

"d'", NP: 18, 11, NS: 18, 15
17: Line, 0, CN:2, VN:2
lBl, W, t, DS:1 1, GT:0, V, nSt
Const: 15 16
p: 0, Times, S: 14 C: 7 Fa: 0

18: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, I, nSt
Const: 12, Val: 0.419751133804922 4.25765549093156

19: Refl, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 18 13

20: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, I, nSt
Const: 12, Val: -6.21975113380492 1.67567784240177

21: Refl, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 20 13

"M", NP: -74, 47, NS: 16, 12
22: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 13 14
p: 0, Times, S: 12 C: 9 Fa: 0

"I", NP: -53, 118, NS: 10, 12
23: Int, 256, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 13 10
p: 0, Times, S: 12 C: 6 Fa: 0

24: Angle, 0, CN:3, VN:5
V, W, t, DS:1 1, GT:24, V, nSt
Const: 23 22 5, Val: 1, 2

25: AngVal, 0, CN:1, VN:1
V, W, NbD:-2, FD, ¡, GT:0, V, nSt
Const: 24, Val: -39 67 1.38535, nA, P
p: 0, Geneva, S: 9 C: 5 Fa: 0

26: Text, 0, CN:2, VN:3
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Const: 24 25, Val: -40 67 0, A, nP, TP: -1.33_, -2.23_, TS: 1.33_, -0.4
""#"
p: 0, Geneva, S: 9 C: 6 Fa: 0

27: Seg, 0, CN:2, VN:2
Gr, W, t, DS:1 1, GT:0, V, nSt
Const: 9 23

28: Angle, 0, CN:3, VN:5
V, W, t, DS:1 1, GT:24, V, nSt
Const: 23 3 9, Val: 1, 3.4

29: AngVal, 0, CN:1, VN:1
V, W, NbD:-2, FD, ¡, GT:0, V, nSt
Const: 28, Val: -69 -31 0.185442, nA, P
p: 0, Geneva, S: 9 C: 5 Fa: 0

30: Text, 0, CN:2, VN:3
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Const: 28 29, Val: -70 -31 0, A, nP, TP: -2.33_, 1.03_, TS: 1.33_, -0.4
""#"
p: 0, Geneva, S: 9 C: 6 Fa: 0

31: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: -54 -167 0, A, nP, TP: -1.8, 5.56_, TS: 12.63_, -0.5
"Tangente au cercle circonscrit et bissectrice du triangle"
p: 0, Times, S: 14 C: 5 Fa: 1

32: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: 102 -132 0, A, nP, TP: 3.4, 4.4, TS: 7.73_, -2.73_
"ABC un triangle. t la tangente en A au cercle circonscrit, Æ = (AM) la bissectrice intŽrieure de l'angle A. On note d la droite (BC) et d' le symŽtrique de d par rapport ˆ la bissectrice Æ.

ThŽorme : Alors  d' // t."
p: 0, Times, S: 12 C: 6 Fa: 0, p: 16, Symbol, S: 12 C: 10 Fa: 0, p: 18, Times, S: 12 C: 6 Fa: 0, p: 59, Times, S: 12 C: 9 Fa: 0, p: 60, Times, S: 12 C: 6 Fa: 0, p: 61, Times, S: 12 C: 9 Fa: 0, p: 67, Times, S: 12 C: 6 Fa: 0, p: 116, Times, S: 12 C: 4 Fa: 0, p: 117, Times, S: 12 C: 6 Fa: 0, p: 128, Times, S: 12 C: 4 Fa: 0, p: 132, Times, S: 12 C: 6 Fa: 0, p: 136, Times, S: 12 C: 7 Fa: 0, p: 138, Times, S: 12 C: 6 Fa: 0, p: 186, Times, S: 12 C: 9 Fa: 0, p: 188, Times, S: 12 C: 6 Fa: 0, p: 191, Times, S: 12 C: 4 Fa: 1, p: 209, Times, S: 12 C: 7 Fa: 1, p: 211, Times, S: 12 C: 4 Fa: 1, p: 215, Symbol, S: 12 C: 10 Fa: 1, p: 216, Times, S: 12 C: 4 Fa: 1

33: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: 102 -30 0, A, nP, TP: 3.4, 1, TS: 8.06_, -7.13_
"Preuve

Soit I l'autre intersection de la bissectrice (AM) avec le cercle circonscrit. Par les angles inscrits, on a les ŽgalitŽs d'angles IBC = IAC = BAI = BCI. Donc BICest  isocle en I. Autrement dit I et O sont sur la mŽdiatrice de [BC].

Il en rŽsulte donc, en angles de droites, que :
 (AM, BC) + (IO, IM) = ¹/2
Or (IO, IM) = (AM, AO) car AIO isocle en O.

Par ailleurs, on a (d', AM) = (AM, BC) par hypothse. On peut donc Žcrire :
(d', t) = (d', AM) + (AM, t)
           =  (AM, BC) + (AM, AO) + (AO, t)
           =  ¹/2 + ¹/2 = 0 [¹]. Cqfd."
p: 0, Times, S: 12 C: 10 Fa: 4, p: 7, Times, S: 12 C: 10 Fa: 0, p: 445, Symbol, S: 12 C: 10 Fa: 0, p: 446, Times, S: 12 C: 10 Fa: 0, p: 465, Symbol, S: 12 C: 10 Fa: 0, p: 467, Times, S: 12 C: 10 Fa: 0, p: 509, Symbol, S: 12 C: 10 Fa: 0, p: 511, Times, S: 12 C: 10 Fa: 0