Mouvement\;de\;surfaces\;\left(  Annexe \right) 

On\;désigne\;par:\
*\;A_x:\;la\;somme\;des\;aires\;de\;deux\;carrés\
*\;A_1(x):\;la\;fonction\;qui\;à\;x∈\left [0;\,1 \; \right ] \;associe\;A_x\
\;\;\;\;\;\;\;On \;montre\;que\;A_1(x)=2x^{2}-2x+1\
*\;V:\;le\;point\;de\;coordonnées\;\left(  x\;,\;A_x \right) \
*\;\Delta:\;la\; droite\;d'équation:y=\frac{3}{4}

Δ:y=\frac{3}{4}

La \;somme\; des\; aires \;des\ carrés\
\;\;\;\;\;\;\;\;\;\; est \;inférieure \;à\;\frac{3}{4}

La \;somme\; des\; aires \;des\ carrés\
\;\;\; \;\;\;\;\;\;est \;supérieure \;à\;\frac{3}{4}

Conjecture:\
*\;1^{ère}\;méthode:\;d'après\;la\;courbe\;\mathscr{C}_{A_1}\
 \;\;\;\;\;\;\;\;\;\;\;\; A_x≤\frac{3}{4}\;\;\;si\;\;\;x∈\left [0.146\;,\;0.854  \right ]\
*\;2^{ème} \;méthode:\;d'après \;le\;calcul\;formel\
\;\;\;A_1(x)≤\frac{3}{4}\;si\;x∈\left [ \frac{-\sqrt{2}+2}{4}\;,\;\frac{\sqrt{2}+2}{4} \right ] 

A

E

B

H

C

G

D

I

J

\mathscr{C}_{A_1}

\Delta:y=0.75

B = (1, 1)
A = (0, 1)
C = (1, 0)
D = (0, 0)
poly1, a, b, c_2, d = Polygon(A, B, C, D)
c = 0.36 (slider from 0 to 1, step 0.0010)
k = Circle(A, c)
J = Intersect(k, d, 1)
I: (c, 1 - c)
f = OrthogonalLine(I, a)
E = Intersect(f, a)
e = Segment(A, C)
g = OrthogonalLine(I, d)
poly2, i, j, l, m = Polygon(J, I, E, A)
G = Intersect(f, c_2)
H = Intersect(g, b)
poly3, i_1, h, c_1, g_1 = Polygon(I, H, C, G)
A_x: poly2 + poly3
Dif: A_x - 0.75
V: (c, A_x)
p = OrthogonalLine(V, yAxis)
q = OrthogonalLine(V, xAxis)
F = Intersect(p, yAxis)
K = Intersect(q, xAxis)
texte4: "x=" + (LaTeX[c]) + ""
texte5: "A_x=" + (LaTeX[A_x]) + ""
L = Point(Δ)
n = Segment(V, F)
texte7: "V\left( " + (LaTeX[c]) + "\;, \;" + (LaTeX[A_x]) + "  \right) "
A167 = 0.55
B167 = 0.505
C167 = -0.245
A168 = 0.553
B168 = 0.505618
C168 = -0.244382
A169 = 0.557
B169 = 0.506498
C169 = -0.243502
A170 = 0.56
B170 = 0.5072000000000001
C170 = -0.2427999999999999
A171 = 0.563
B171 = 0.507938
C171 = -0.242062
A172 = 0.567
B172 = 0.5089779999999999
C172 = -0.24102200000000007
A173 = 0.57
B173 = 0.5098
C173 = -0.24019999999999997
A174 = 0.573
B174 = 0.5106580000000001
C174 = -0.23934199999999994
A175 = 0.577
B175 = 0.5118579999999999
C175 = -0.23814200000000008
A176 = 0.58
B176 = 0.5127999999999999
C176 = -0.23720000000000008
A177 = 0.583
B177 = 0.5137780000000001
C177 = -0.23622199999999993
A178 = 0.587
B178 = 0.515138
C178 = -0.23486200000000002
A179 = 0.59
B179 = 0.5162
C179 = -0.2338
A180 = 0.593
B180 = 0.517298
C180 = -0.23270199999999996
A181 = 0.597
B181 = 0.518818
C181 = -0.231182
A182 = 0.6
B182 = 0.52
C182 = -0.22999999999999998
A183 = 0.603
B183 = 0.521218
C183 = -0.22878200000000004
A184 = 0.607
B184 = 0.5228980000000001
C184 = -0.22710199999999992
A185 = 0.61
B185 = 0.5242
C185 = -0.2258
A186 = 0.613
B186 = 0.5255380000000001
C186 = -0.22446199999999994
A187 = 0.617
B187 = 0.527378
C187 = -0.222622
A188 = 0.62
B188 = 0.5287999999999999
C188 = -0.22120000000000006
A189 = 0.623
B189 = 0.530258
C189 = -0.219742
A190 = 0.627
B190 = 0.532258
C190 = -0.217742
A191 = 0.63
B191 = 0.5338
C191 = -0.21619999999999995
A192 = 0.633
B192 = 0.535378
C192 = -0.21462199999999998
A193 = 0.637
B193 = 0.5375380000000001
C193 = -0.21246199999999993
A194 = 0.64
B194 = 0.5392
C194 = -0.2108
A195 = 0.643
B195 = 0.540898
C195 = -0.209102
A196 = 0.647
B196 = 0.543218
C196 = -0.20678200000000002
A197 = 0.65
B197 = 0.545
C197 = -0.20499999999999996
A198 = 0.653
B198 = 0.546818
C198 = -0.20318199999999997
A199 = 0.657
B199 = 0.5492980000000001
C199 = -0.20070199999999994
A200 = 0.66
B200 = 0.5512
C200 = -0.19879999999999998
A201 = 0.663
B201 = 0.553138
C201 = -0.19686199999999998
A202 = 0.667
B202 = 0.5557780000000001
C202 = -0.1942219999999999
A203 = 0.67
B203 = 0.5578000000000001
C203 = -0.19219999999999993
A204 = 0.673
B204 = 0.559858
C204 = -0.19014200000000003
A205 = 0.677
B205 = 0.562658
C205 = -0.187342
A206 = 0.68
B206 = 0.5648000000000001
C206 = -0.18519999999999992
A207 = 0.683
B207 = 0.5669780000000001
C207 = -0.1830219999999999
A208 = 0.687
B208 = 0.5699380000000001
C208 = -0.18006199999999994
A209 = 0.69
B209 = 0.5722
C209 = -0.17779999999999996
A210 = 0.693
B210 = 0.574498
C210 = -0.17550200000000005
A211 = 0.697
B211 = 0.577618
C211 = -0.17238200000000004
A212 = 0.7
B212 = 0.58
C212 = -0.17000000000000004
A213 = 0.703
B213 = 0.5824179999999999
C213 = -0.16758200000000012
A214 = 0.707
B214 = 0.5856979999999999
C214 = -0.16430200000000006
A215 = 0.71
B215 = 0.5882000000000001
C215 = -0.16179999999999994
A216 = 0.713
B216 = 0.590738
C216 = -0.15926200000000001
A217 = 0.717
B217 = 0.5941780000000001
C217 = -0.1558219999999999
A218 = 0.72
B218 = 0.5968
C218 = -0.1532
A219 = 0.723
B219 = 0.599458
C219 = -0.15054199999999995
A220 = 0.727
B220 = 0.6030580000000001
C220 = -0.1469419999999999
A221 = 0.73
B221 = 0.6057999999999999
C221 = -0.1442000000000001
A222 = 0.733
B222 = 0.6085780000000001
C222 = -0.14142199999999994
A223 = 0.737
B223 = 0.612338
C223 = -0.13766199999999995
A224 = 0.74
B224 = 0.6152
C224 = -0.13480000000000003
A225 = 0.743
B225 = 0.618098
C225 = -0.13190199999999996
A226 = 0.747
B226 = 0.622018
C226 = -0.12798200000000004
A227 = 0.75
B227 = 0.625
C227 = -0.125
A228 = 0.753
B228 = 0.628018
C228 = -0.12198200000000003
A229 = 0.757
B229 = 0.632098
C229 = -0.11790199999999995
A230 = 0.76
B230 = 0.6352
C230 = -0.11480000000000001
A231 = 0.763
B231 = 0.6383380000000001
C231 = -0.11166199999999993
A232 = 0.767
B232 = 0.6425780000000001
C232 = -0.1074219999999999
A233 = 0.77
B233 = 0.6457999999999999
C233 = -0.10420000000000007
A234 = 0.773
B234 = 0.6490579999999999
C234 = -0.10094200000000009
A235 = 0.777
B235 = 0.6534580000000001
C235 = -0.0965419999999999
A236 = 0.78
B236 = 0.6568
C236 = -0.09319999999999995
A237 = 0.783
B237 = 0.6601779999999999
C237 = -0.08982200000000007
A238 = 0.787
B238 = 0.664738
C238 = -0.08526199999999995
A239 = 0.79
B239 = 0.6682000000000001
C239 = -0.08179999999999987
A240 = 0.793
B240 = 0.6716979999999999
C240 = -0.0783020000000001
A241 = 0.797
B241 = 0.6764180000000002
C241 = -0.07358199999999981
A242 = 0.8
B242 = 0.6800000000000002
C242 = -0.06999999999999984
A243 = 0.803
B243 = 0.6836180000000001
C243 = -0.06638199999999994
A244 = 0.807
B244 = 0.688498
C244 = -0.061501999999999946
A245 = 0.81
B245 = 0.6922000000000001
C245 = -0.05779999999999985
A246 = 0.813
B246 = 0.695938
C246 = -0.054062000000000054
A247 = 0.817
B247 = 0.7009779999999999
C247 = -0.04902200000000012
A248 = 0.82
B248 = 0.7047999999999999
C248 = -0.04520000000000013
A249 = 0.823
B249 = 0.708658
C249 = -0.04134199999999999
A250 = 0.827
B250 = 0.7138579999999999
C250 = -0.03614200000000012
A251 = 0.83
B251 = 0.7178
C251 = -0.032200000000000006
A252 = 0.833
B252 = 0.721778
C252 = -0.02822199999999997
A253 = 0.837
B253 = 0.7271380000000001
C253 = -0.022861999999999938
A254 = 0.84
B254 = 0.7312000000000001
C254 = -0.018799999999999928
A255 = 0.843
B255 = 0.735298
C255 = -0.014701999999999993
A256 = 0.847
B256 = 0.740818
C256 = -0.009182000000000023
A257 = 0.85
B257 = 0.7449999999999999
C257 = -0.0050000000000001155
A258 = 0.853
B258 = 0.7492179999999999
C258 = -7.820000000000604E-4
A259 = 0.857
B259 = 0.7548980000000001
C259 = 0.004898000000000069
A260 = 0.86
B260 = 0.7592000000000001
C260 = 0.009200000000000097
A261 = 0.863
B261 = 0.763538
C261 = 0.01353800000000005
A262 = 0.867
B262 = 0.7693779999999999
C262 = 0.019377999999999895
A263 = 0.87
B263 = 0.7738
C263 = 0.023800000000000043
A264 = 0.873
B264 = 0.7782580000000001
C264 = 0.028258000000000116
A265 = 0.877
B265 = 0.7842579999999999
C265 = 0.0342579999999999
A266 = 0.88
B266 = 0.7888
C266 = 0.038799999999999946
A267 = 0.883
B267 = 0.7933779999999999
C267 = 0.04337799999999992
A268 = 0.887
B268 = 0.7995380000000001
C268 = 0.04953800000000008
A269 = 0.89
B269 = 0.8042
C269 = 0.054200000000000026
A270 = 0.893
B270 = 0.8088980000000001
C270 = 0.05889800000000012
A271 = 0.897
B271 = 0.815218
C271 = 0.065218
A272 = 0.9
B272 = 0.8200000000000001
C272 = 0.07000000000000006
A273 = 0.903
B273 = 0.824818
C273 = 0.07481800000000005
A274 = 0.907
B274 = 0.8312980000000001
C274 = 0.08129800000000009
A275 = 0.91
B275 = 0.8362
C275 = 0.08620000000000005
A276 = 0.913
B276 = 0.8411379999999999
C276 = 0.09113799999999994
A277 = 0.917
B277 = 0.8477780000000001
C277 = 0.09777800000000014
A278 = 0.92
B278 = 0.8528
C278 = 0.1028
A279 = 0.923
B279 = 0.857858
C279 = 0.10785800000000001
A280 = 0.927
B280 = 0.8646580000000001
C280 = 0.11465800000000015
A281 = 0.93
B281 = 0.8698000000000001
C281 = 0.11980000000000013
A282 = 0.933
B282 = 0.874978
C282 = 0.12497800000000003
A283 = 0.937
B283 = 0.8819380000000001
C283 = 0.1319380000000001
A284 = 0.94
B284 = 0.8872
C284 = 0.1372
A285 = 0.943
B285 = 0.892498
C285 = 0.142498
A286 = 0.947
B286 = 0.899618
C286 = 0.14961800000000003
A287 = 0.95
B287 = 0.9049999999999998
C287 = 0.1549999999999998
A288 = 0.953
B288 = 0.910418
C288 = 0.16041799999999995
A289 = 0.957
B289 = 0.9176979999999999
C289 = 0.1676979999999999
A290 = 0.96
B290 = 0.9232
C290 = 0.17320000000000002
A291 = 0.963
B291 = 0.9287379999999998
C291 = 0.17873799999999984
A292 = 0.967
B292 = 0.936178
C292 = 0.18617799999999995
A293 = 0.97
B293 = 0.9418
C293 = 0.19179999999999997
A294 = 0.973
B294 = 0.9474579999999999
C294 = 0.1974579999999999
A295 = 0.977
B295 = 0.955058
C295 = 0.20505799999999996
A296 = 0.98
B296 = 0.9607999999999999
C296 = 0.21079999999999988
A297 = 0.983
B297 = 0.9665779999999999
C297 = 0.21657799999999994
A298 = 0.987
B298 = 0.9743379999999999
C298 = 0.22433799999999993
A299 = 0.99
B299 = 0.9802
C299 = 0.23019999999999996
A300 = 0.993
B300 = 0.9860979999999999
C300 = 0.23609799999999992
A301 = 0.997
B301 = 0.9940180000000001
C301 = 0.24401800000000007
A302 = 1
B302 = 1
C302 = 0.25
A303 = 1
B303 = 1
C303 = 0.25
f_1(x) = 2 * x^(2) - 2 * x + 1
A_1 = Function(f_1, 0, 1)
M, N = Intersect(A_1, Δ)
A_2 = Function(f_1, x(M), x(N))
s: x(N)
t = OrthogonalLine(N, xAxis)
O = Intersect(s, xAxis)
P = Intersect(t, xAxis)
a_1 = Segment(M, O)
b_1 = Segment(N, P)
d_1 = Segment(O, P)
r: x(M)
r': x(M)
texte20: "x=" + (LaTeX[r']) + ""
texte21: "x=" + (LaTeX[s]) + ""
e_1 = OrthogonalLine(M, xAxis)
Q = Intersect(e_1, xAxis)
h_1 = Segment(M, Q)
texte22: "x=" + (LaTeX[r]) + ""
j_1 = Segment(Q, P)
k_1 = Segment(V, K)
texte2: "Pour\;x=c=" + (LaTeX[c]) + ":\\
on\;a:A_x=" + (LaTeX[A_x]) + ""