V

P

B

O

On\;considère\;le\;point:\
\;\;\;\;\;\;\;\;\;\;V\left(x\;,\;S(x)   \right)

Conjectures:\
1^{ère} \;méthode:d'après\;les\;traces\;du\;point\;V\;on\;a:\
\;\;\;\;\;\;\;\;\;S(x)=27\;\;\;\;\; pour\;\; \;\;x=3\
2^{ème }\;méthode:\;d'après\;le\;tableur\
*\;Ouvrir\;le\;menu\;''Affichage''\*\;désactiver\;l'option\;Graphique\;2 \;\*\;activer\;l'option\;''Tableur''   .\
la\;ligne\;151\;de\;ce\;tableur\;confirme\;les\;mêmes\;résultats

Le\; pas\;de\;la\;colonne\; OP\;dans\;le\;tableur\
est\;10^{-2}\;c'est\;l'incrément\;du \;curseur\;OP=x

C

5

-5

0

5

A

10

N

M

Q

Le\; plus\; grand\; rectangle

Commande\; de\;la\;distance \;OP=x

On\;considère\;le\;point\;V(x\;,\;S(x))

O = (-2.7916479262865743, -3.992634136682949)
c_1 = Circle(O, 1)
a: y = y(O)
B = Intersect(c_1, a, 2)
O' = Mirror(O, B)
B' = Mirror(B, O')
O'' = Mirror(O', B')
B'' = Mirror(B', O'')
O''' = Mirror(O'', B'')
e = Segment(O, O''')
OP = 3 (slider from 0 to 6, step 0.01)
p = Circle(O, OP)
r = 1
B''' = Mirror(B'', O''')
D = Mirror(B''', O)
E = Mirror(O''', O)
D' = Mirror(D, E)
E' = Mirror(E, D')
D'' = Mirror(D', E')
E'' = Mirror(E', D'')
D''' = Mirror(D'', E'')
b = Segment(D, B''')
d = OrthogonalLine(O, b)
F = Intersect(c_1, d, 2)
O'_1 = Mirror(O, F)
F' = Mirror(F, O'_1)
G = Mirror(O'_1, F')
F'' = Mirror(F', G)
G' = Mirror(G, F'')
F''' = Mirror(F'', G')
G'' = Mirror(G', F''')
C = Mirror(F''', G'')
G''' = Mirror(G'', C)
poly1, g, h, i = Polygon(E, C, O''')
A = Intersect(p, e, 2)
f = OrthogonalLine(A, b)
H = Intersect(f, h)
j = OrthogonalLine(H, f)
I = Intersect(j, g)
k = OrthogonalLine(I, b)
J = Intersect(k, b)
poly2, i_1, h_1, a_1, j_1 = Polygon(I, H, A, J)
texte13: "x=OP=  " + (LaTeX[OP]) + "\\
S(x)=" + (LaTeX[poly2]) + ""
V: (OP, poly2)
texte6: "S(x)=" + (LaTeX[poly2]) + ""
l = OrthogonalLine(V, xAxis)
m = OrthogonalLine(V, yAxis)
K = Intersect(m, yAxis)
L = Intersect(l, xAxis)
n = Segment(V, K)
q = Segment(V, L)
texte4: "x=" + (LaTeX[OP]) + ""
A1 = Name(OP)
B1 = Name(poly2)
A2 = 0.02
B2 = 0.358799999998876
A3 = 0.04
B3 = 0.7151999999995269
A4 = 0.06
B4 = 1.0691999999994604
A5 = 0.08
B5 = 1.4207999999997512
A6 = 0.1
B6 = 1.7699999999996194
A7 = 0.12
B7 = 2.116799999999987
A8 = 0.14
B8 = 2.4611999999999075
A9 = 0.16
B9 = 2.803199999999931
A10 = 0.18
B10 = 3.142800000000026
A11 = 0.2
B11 = 3.4799999999998708
A12 = 0.22
B12 = 3.8147999999999733
A13 = 0.24
B13 = 4.147199999999941
A14 = 0.26
B14 = 4.477199999999911
A15 = 0.28
B15 = 4.8048000000000055
A16 = 0.3
B16 = 5.129999999999969
A17 = 0.32
B17 = 5.45279999999995
A18 = 0.34
B18 = 5.773199999999978
A19 = 0.36
B19 = 6.091199999999951
A20 = 0.38
B20 = 6.4067999999999055
A21 = 0.4
B21 = 6.719999999999969
A22 = 0.42
B22 = 7.03079999999999
A23 = 0.44
B23 = 7.339199999999986
A24 = 0.46
B24 = 7.645199999999986
A25 = 0.48
B25 = 7.948799999999971
A26 = 0.5
B26 = 8.249999999999972
A27 = 0.52
B27 = 8.5488
A28 = 0.54
B28 = 8.845200000000016
A29 = 0.56
B29 = 9.139199999999992
A30 = 0.58
B30 = 9.430799999999948
A31 = 0.6
B31 = 9.719999999999978
A32 = 0.62
B32 = 10.006799999999977
A33 = 0.64
B33 = 10.291199999999957
A34 = 0.66
B34 = 10.573199999999982
A35 = 0.68
B35 = 10.852800000000007
A36 = 0.7
B36 = 11.13
A37 = 0.72
B37 = 11.40479999999998
A38 = 0.74
B38 = 11.67719999999998
A39 = 0.76
B39 = 11.947199999999988
A40 = 0.78
B40 = 12.214799999999984
A41 = 0.8
B41 = 12.479999999999995
A42 = 0.82
B42 = 12.742799999999985
A43 = 0.84
B43 = 13.003199999999998
A44 = 0.86
B44 = 13.261199999999977
A45 = 0.88
B45 = 13.516799999999996
A46 = 0.9
B46 = 13.770000000000007
A47 = 0.92
B47 = 14.020799999999987
A48 = 0.94
B48 = 14.2692
A49 = 0.96
B49 = 14.51519999999999
A50 = 0.98
B50 = 14.758799999999987
A51 = 1
B51 = 15.000000000000004
A52 = 1.02
B52 = 15.238799999999975
A53 = 1.04
B53 = 15.475200000000001
A54 = 1.06
B54 = 15.709199999999992
A55 = 1.08
B55 = 15.940799999999992
A56 = 1.1
B56 = 16.169999999999998
A57 = 1.12
B57 = 16.3968
A58 = 1.14
B58 = 16.621199999999988
A59 = 1.16
B59 = 16.84320000000001
A60 = 1.18
B60 = 17.062800000000003
A61 = 1.2
B61 = 17.27999999999998
A62 = 1.22
B62 = 17.494799999999998
A63 = 1.24
B63 = 17.7072
A64 = 1.26
B64 = 17.91720000000001
A65 = 1.28
B65 = 18.1248
A66 = 1.3
B66 = 18.330000000000005
A67 = 1.32
B67 = 18.532799999999995
A68 = 1.34
B68 = 18.733200000000007
A69 = 1.36
B69 = 18.93120000000001
A70 = 1.38
B70 = 19.126799999999996
A71 = 1.4
B71 = 19.32
A72 = 1.42
B72 = 19.510800000000007
A73 = 1.44
B73 = 19.699199999999998
A74 = 1.46
B74 = 19.885199999999994
A75 = 1.48
B75 = 20.068799999999996
A76 = 1.5
B76 = 20.25
A77 = 1.52
B77 = 20.4288
A78 = 1.54
B78 = 20.605200000000007
A79 = 1.56
B79 = 20.7792
A80 = 1.58
B80 = 20.950800000000005
A81 = 1.6
B81 = 21.120000000000005
A82 = 1.62
B82 = 21.28680000000001
A83 = 1.64
B83 = 21.4512
A84 = 1.66
B84 = 21.613200000000006
A85 = 1.68
B85 = 21.772800000000004
A86 = 1.7
B86 = 21.930000000000007
A87 = 1.72
B87 = 22.084799999999998
A88 = 1.74
B88 = 22.2372
A89 = 1.76
B89 = 22.387200000000004
A90 = 1.78
B90 = 22.534799999999997
A91 = 1.8
B91 = 22.680000000000003
A92 = 1.82
B92 = 22.8228
A93 = 1.84
B93 = 22.963200000000008
A94 = 1.86
B94 = 23.101200000000013
A95 = 1.88
B95 = 23.236800000000002
A96 = 1.9
B96 = 23.369999999999997
A97 = 1.92
B97 = 23.500800000000005
A98 = 1.94
B98 = 23.629200000000004
A99 = 1.96
B99 = 23.75520000000001
A100 = 1.98
B100 = 23.878800000000005
A101 = 2
B101 = 24.000000000000007
A102 = 2.02
B102 = 24.1188
A103 = 2.04
B103 = 24.23520000000001
A104 = 2.06
B104 = 24.349200000000003
A105 = 2.08
B105 = 24.460800000000006
A106 = 2.1
B106 = 24.57
A107 = 2.12
B107 = 24.676800000000007
A108 = 2.14
B108 = 24.781200000000005
A109 = 2.16
B109 = 24.88320000000001
A110 = 2.18
B110 = 24.982800000000008
A111 = 2.2
B111 = 25.080000000000005
A112 = 2.22
B112 = 25.174800000000005
A113 = 2.24
B113 = 25.26720000000001
A114 = 2.26
B114 = 25.357200000000006
A115 = 2.28
B115 = 25.444800000000008
A116 = 2.3
B116 = 25.53000000000001
A117 = 2.32
B117 = 25.612800000000007
A118 = 2.34
B118 = 25.693200000000008
A119 = 2.36
B119 = 25.771200000000007
A120 = 2.38
B120 = 25.84680000000001
A121 = 2.4
B121 = 25.92
A122 = 2.42
B122 = 25.99080000000001
A123 = 2.44
B123 = 26.05920000000001
A124 = 2.46
B124 = 26.12520000000001
A125 = 2.48
B125 = 26.188800000000008
A126 = 2.5
B126 = 26.25000000000001
A127 = 2.52
B127 = 26.30880000000001
A128 = 2.54
B128 = 26.36520000000001
A129 = 2.56
B129 = 26.419200000000004
A130 = 2.58
B130 = 26.470800000000008
A131 = 2.6
B131 = 26.52000000000001
A132 = 2.62
B132 = 26.566800000000008
A133 = 2.64
B133 = 26.611200000000004
A134 = 2.66
B134 = 26.653200000000012
A135 = 2.68
B135 = 26.692800000000005
A136 = 2.7
B136 = 26.730000000000008
A137 = 2.72
B137 = 26.764800000000008
A138 = 2.74
B138 = 26.797200000000004
A139 = 2.76
B139 = 26.827200000000005
A140 = 2.78
B140 = 26.854800000000008
A141 = 2.8
B141 = 26.880000000000003
A142 = 2.82
B142 = 26.902800000000006
A143 = 2.84
B143 = 26.92320000000001
A144 = 2.86
B144 = 26.94120000000001
A145 = 2.88
B145 = 26.95680000000001
A146 = 2.9
B146 = 26.970000000000006
A147 = 2.92
B147 = 26.98080000000001
A148 = 2.94
B148 = 26.989200000000007
A149 = 2.96
B149 = 26.995200000000008
A150 = 2.98
B150 = 26.998800000000006
A151 = 3
B151 = 27.00000000000001
A152 = 3.02
B152 = 26.998800000000006
A153 = 3.04
B153 = 26.995200000000008
A154 = 3.06
B154 = 26.989200000000007
A155 = 3.08
B155 = 26.980800000000013
A156 = 3.1
B156 = 26.970000000000006
A157 = 3.12
B157 = 26.95680000000001
A158 = 3.14
B158 = 26.94120000000001
A159 = 3.16
B159 = 26.92320000000001
A160 = 3.18
B160 = 26.90280000000001
A161 = 3.2
B161 = 26.88000000000001
A162 = 3.22
B162 = 26.854800000000008
A163 = 3.24
B163 = 26.827200000000005
A164 = 3.26
B164 = 26.797200000000007
A165 = 3.28
B165 = 26.76480000000001
A166 = 3.3
B166 = 26.730000000000004
A167 = 3.32
B167 = 26.69280000000001
A168 = 3.34
B168 = 26.653200000000005
A169 = 3.36
B169 = 26.611200000000004
A170 = 3.38
B170 = 26.566800000000008
A171 = 3.4
B171 = 26.520000000000003
A172 = 3.42
B172 = 26.470800000000008
A173 = 3.44
B173 = 26.41920000000001
A174 = 3.46
B174 = 26.36520000000001
A175 = 3.48
B175 = 26.30880000000001
A176 = 3.5
B176 = 26.250000000000007
A177 = 3.52
B177 = 26.188800000000008
A178 = 3.54
B178 = 26.125200000000007
A179 = 3.56
B179 = 26.059200000000004
A180 = 3.58
B180 = 25.990800000000007
A181 = 3.6
B181 = 25.92000000000001
A182 = 3.62
B182 = 25.84680000000001
A183 = 3.64
B183 = 25.771200000000007
A184 = 3.66
B184 = 25.693200000000004
A185 = 3.68
B185 = 25.612800000000007
A186 = 3.7
B186 = 25.53
A187 = 3.72
B187 = 25.444800000000008
A188 = 3.74
B188 = 25.357200000000006
A189 = 3.76
B189 = 25.26720000000001
A190 = 3.78
B190 = 25.174800000000005
A191 = 3.8
B191 = 25.080000000000005
A192 = 3.82
B192 = 24.982800000000008
A193 = 3.84
B193 = 24.883200000000006
A194 = 3.86
B194 = 24.781200000000005
A195 = 3.88
B195 = 24.676800000000007
A196 = 3.9
B196 = 24.570000000000007
A197 = 3.92
B197 = 24.460800000000006
A198 = 3.94
B198 = 24.34920000000001
A199 = 3.96
B199 = 24.235200000000006
A200 = 3.98
B200 = 24.118800000000007
A201 = 4
B201 = 24.000000000000007
A202 = 4.02
B202 = 23.87880000000001
A203 = 4.04
B203 = 23.755200000000006
A204 = 4.06
B204 = 23.62920000000001
A205 = 4.08
B205 = 23.500800000000005
A206 = 4.1
B206 = 23.370000000000005
A207 = 4.12
B207 = 23.236800000000006
A208 = 4.14
B208 = 23.10120000000001
A209 = 4.16
B209 = 22.963200000000004
A210 = 4.18
B210 = 22.822800000000008
A211 = 4.2
B211 = 22.680000000000007
A212 = 4.22
B212 = 22.534800000000008
A213 = 4.24
B213 = 22.387200000000004
A214 = 4.26
B214 = 22.23720000000001
A215 = 4.28
B215 = 22.0848
A216 = 4.3
B216 = 21.930000000000007
A217 = 4.32
B217 = 21.772800000000004
A218 = 4.34
B218 = 21.613200000000006
A219 = 4.36
B219 = 21.4512
A220 = 4.38
B220 = 21.286800000000003
A221 = 4.4
B221 = 21.120000000000005
A222 = 4.42
B222 = 20.9508
A223 = 4.44
B223 = 20.779200000000003
A224 = 4.46
B224 = 20.605200000000004
A225 = 4.48
B225 = 20.428800000000003
A226 = 4.5
B226 = 20.25
A227 = 4.52
B227 = 20.068800000000007
A228 = 4.54
B228 = 19.885200000000005
A229 = 4.56
B229 = 19.699200000000005
A230 = 4.58
B230 = 19.510800000000003
A231 = 4.6
B231 = 19.320000000000007
A232 = 4.62
B232 = 19.126800000000003
A233 = 4.64
B233 = 18.931200000000008
A234 = 4.66
B234 = 18.7332
A235 = 4.68
B235 = 18.53280000000001
A236 = 4.7
B236 = 18.330000000000005
A237 = 4.72
B237 = 18.124799999999997
A238 = 4.74
B238 = 17.917199999999998
A239 = 4.76
B239 = 17.707200000000004
A240 = 4.78
B240 = 17.4948
A241 = 4.8
B241 = 17.280000000000005
A242 = 4.82
B242 = 17.062800000000006
A243 = 4.84
B243 = 16.843200000000007
A244 = 4.86
B244 = 16.6212
A245 = 4.88
B245 = 16.396800000000002
A246 = 4.9
B246 = 16.169999999999995
A247 = 4.92
B247 = 15.940800000000001
A248 = 4.94
B248 = 15.7092
A249 = 4.96
B249 = 15.475200000000008
A250 = 4.98
B250 = 15.238799999999994
A251 = 5
B251 = 14.999999999999995
A252 = 5.02
B252 = 14.758800000000008
A253 = 5.04
B253 = 14.515199999999991
A254 = 5.06
B254 = 14.269200000000009
A255 = 5.08
B255 = 14.020799999999998
A256 = 5.1
B256 = 13.769999999999996
A257 = 5.12
B257 = 13.516799999999996
A258 = 5.14
B258 = 13.261200000000006
A259 = 5.16
B259 = 13.003200000000007
A260 = 5.18
B260 = 12.742800000000013
A261 = 5.2
B261 = 12.480000000000011
A262 = 5.22
B262 = 12.214799999999997
A263 = 5.24
B263 = 11.947199999999995
A264 = 5.26
B264 = 11.677200000000012
A265 = 5.28
B265 = 11.404800000000005
A266 = 5.3
B266 = 11.13000000000001
A267 = 5.32
B267 = 10.852800000000002
A268 = 5.34
B268 = 10.573199999999986
A269 = 5.36
B269 = 10.291199999999993
A270 = 5.38
B270 = 10.00679999999999
A271 = 5.4
B271 = 9.719999999999994
A272 = 5.42
B272 = 9.430800000000003
A273 = 5.44
B273 = 9.139199999999992
A274 = 5.46
B274 = 8.845199999999988
A275 = 5.48
B275 = 8.548800000000002
A276 = 5.5
B276 = 8.25
A277 = 5.52
B277 = 7.948800000000009
A278 = 5.54
B278 = 7.645199999999985
A279 = 5.56
B279 = 7.339199999999993
A280 = 5.58
B280 = 7.030799999999988
A281 = 5.6
B281 = 6.720000000000004
A282 = 5.62
B282 = 6.4067999999999845
A283 = 5.64
B283 = 6.091199999999992
A284 = 5.66
B284 = 5.7731999999999815
A285 = 5.68
B285 = 5.452799999999992
A286 = 5.7
B286 = 5.129999999999983
A287 = 5.72
B287 = 4.804799999999991
A288 = 5.74
B288 = 4.477200000000001
A289 = 5.76
B289 = 4.147199999999994
A290 = 5.78
B290 = 3.8147999999999964
A291 = 5.8
B291 = 3.479999999999986
A292 = 5.82
B292 = 3.1427999999999736
A293 = 5.84
B293 = 2.803200000000003
A294 = 5.86
B294 = 2.4611999999999994
A295 = 5.88
B295 = 2.1168000000000005
A296 = 5.9
B296 = 1.7699999999999727
A297 = 5.92
B297 = 1.420799999999992
A298 = 5.94
B298 = 1.0691999999999773
A299 = 5.96
B299 = 0.7151999999999845
A300 = 5.98
B300 = 0.3587999999999957
A301 = 6
B301 = 1.266516531290768E-14