x=\frac{\pi}{2}

B

A

I

M

G

F

E

\mathscr{C}

1

2

0

V

Carré\;d'aire\;variable\

On\; définit\; le \;point\;V(\alpha,a)

Conjecture \;concernant \;les \;variations\; de \;a:\
*\;1^{ère}\;méthode:d'après \;les \;traces\; de\; V(\alpha,a)\
a \;est \;croissante \;sur\;]0\;,\frac{\pi}{2}[\;et\;décroissante\
sur\;]\frac{\pi}{2}\;,\pi[\
*\;2^{ème}\;méthode:\;d'après\;le\;tableur\
les\; résultats\; sont\; confirmés:\
voir\; ce\; qui\;précède\;et\;ce\;qui\;succède\
les\;lignes\;91,\;92\;et\;93\;(en\;jaune)

Conjecture \;concernant \;les \;solutions\; de \;a=16:\
*\;1^{ère}\;méthode:d'après \;les \;traces\; de\; V(\alpha,a)\
V\;appartient\;(presque) \;à \;la\;droite\;d'équation\;y=16\
lorsque\;\alpha≈1.047\;rad\;ou\;\alpha≈2.094\;rad\
*\;2^{ème}\;méthode:\;d'après\;le\;tableur\
les\; résultats\; sont\; confirmés:\
voir\; les\; lignes\;62,\;et\;122\;(en\;vert)

T

I = Intersect(a, b)
A = (-8, 2)
B = (-4, 2)
c = Semicircle(A, B)
α = 0.88 rad (slider from 0 to 3.141592653589793, step 0.0010)
M = Rotate(B, α, I)
T = Tangent(M, c)
E = Intersect(b, T)
poly1, d, e, f, g, G, F = Polygon(E, I, 4)
V: (α, poly1)
A1: "x(" + (Name[V] + ")")
B1: "y(" + (Name[V] + ")")
A2 = 0
B2 = 4
A3 = 0.017453292519943295
B3 = 4.0012187171665925
A4 = 0.03490658503988659
B4 = 4.00487784056758
A5 = 0.05235987755982989
B5 = 4.010986301317522
A6 = 0.06981317007977318
B6 = 4.019559036114335
A7 = 0.08726646259971647
B7 = 4.030617064982209
A8 = 0.10471975511965978
B8 = 4.044187601083132
A9 = 0.12217304763960307
B9 = 4.0603041935844795
A10 = 0.13962634015954636
B10 = 4.079006904875382
A11 = 0.15707963267948966
B11 = 4.10034252374767
A12 = 0.17453292519943295
B12 = 4.124364816503052
A13 = 0.19198621771937624
B13 = 4.151134818321897
A14 = 0.20943951023931956
B14 = 4.1807211676359275
A15 = 0.22689280275926285
B15 = 4.213200486694359
A16 = 0.24434609527920614
B16 = 4.2486578120070755
A17 = 0.2617993877991494
B17 = 4.28718707889796
A18 = 0.2792526803190927
B18 = 4.328891665015862
A19 = 0.29670597283903605
B19 = 4.373884998340702
A20 = 0.3141592653589793
B20 = 4.422291236000334
A21 = 0.33161255787892263
B21 = 4.474246021095033
A22 = 0.3490658503988659
B22 = 4.5298973257271795
A23 = 0.3665191429188092
B23 = 4.589406389574682
A24 = 0.3839724354387525
B24 = 4.652948764650405
A25 = 0.4014257279586958
B25 = 4.7207154783847125
A26 = 0.4188790204786391
B26 = 4.792914328886268
A27 = 0.4363323129985824
B27 = 4.869771328215993
A28 = 0.4537856055185257
B28 = 4.951532311796627
A29 = 0.47123889803846897
B29 = 5.038464734730005
A30 = 0.4886921905584123
B30 = 5.130859678871087
A31 = 0.5061454830783556
B31 = 5.229034098089874
A32 = 0.5235987755982988
B32 = 5.333333333333333
A33 = 0.5410520681182421
B33 = 5.44413393399272
A34 = 0.5585053606381855
B34 = 5.561846827822327
A35 = 0.5759586531581288
B35 = 5.6869208884103415
A36 = 0.5934119456780721
B36 = 5.819846957171869
A37 = 0.6108652381980153
B37 = 5.961162386262806
A38 = 0.6283185307179586
B38 = 6.111456180001674
A39 = 0.6457718232379019
B39 = 6.2713748257052835
A40 = 0.6632251157578453
B40 = 6.441628920743966
A41 = 0.6806784082777885
B41 = 6.623000721670283
A42 = 0.6981317007977318
B42 = 6.816352764167387
A43 = 0.7155849933176751
B43 = 7.022637730172708
A44 = 0.7330382858376184
B44 = 7.242909771951644
A45 = 0.7504915783575618
B45 = 7.478337543497331
A46 = 0.767944870877505
B46 = 7.730219239153537
A47 = 0.7853981633974483
B47 = 8.000000000000014
A48 = 0.8028514559173916
B48 = 8.289292123116766
A49 = 0.8203047484373349
B49 = 8.59989860096001
A50 = 0.8377580409572782
B50 = 8.933840632380907
A51 = 0.8552113334772214
B51 = 9.293389889328816
A52 = 0.8726646259971648
B52 = 9.681106501844816
A53 = 0.8901179185171081
B53 = 10.099883948873392
A54 = 0.9075712110370514
B54 = 10.553002327284348
A55 = 0.9250245035569946
B55 = 11.044191834358186
A56 = 0.9424777960769379
B56 = 11.577708763999633
A57 = 0.9599310885968813
B57 = 12.158426916645897
A58 = 0.9773843811168246
B58 = 12.791948101429732
A59 = 0.9948376736367679
B59 = 13.484736427134777
A60 = 1.0122909661567112
B60 = 14.244282419364179
A61 = 1.0297442586766545
B61 = 15.079304781491622
A62 = 1.0471975511965976
B62 = 16.000000000000014
A63 = 1.064650843716541
B63 = 17.018353213199443
A64 = 1.0821041362364843
B64 = 18.148528149816475
A65 = 1.0995574287564276
B65 = 19.40735998527675
A66 = 1.117010721276371
B66 = 20.81498337117928
A67 = 1.1344640137963142
B67 = 22.395639728453574
A68 = 1.1519173063162575
B68 = 24.178724758920257
A69 = 1.1693705988362009
B69 = 26.200161478229482
A70 = 1.1868238913561442
B70 = 28.504219727816775
A71 = 1.2042771838760873
B71 = 31.145956139948833
A72 = 1.2217304763960306
B72 = 34.194528681652145
A73 = 1.239183768915974
B73 = 37.73776328804228
A74 = 1.2566370614359172
B74 = 41.88854381999823
A75 = 1.2740903539558606
B75 = 46.79390740737818
A76 = 1.2915436464758039
B76 = 52.64823801244075
A77 = 1.3089969389957472
B77 = 59.71281292110187
A78 = 1.3264502315156905
B78 = 68.34545478725839
A79 = 1.3439035240356338
B79 = 79.0467329980229
A80 = 1.361356816555577
B80 = 92.53417786804474
A81 = 1.3788101090755203
B81 = 109.8657440929448
A82 = 1.3962634015954636
B82 = 132.65374991010478
A83 = 1.413716694115407
B83 = 163.4538327562462
A84 = 1.4311699866353502
B84 = 206.51394514488703
A85 = 1.4486232791552935
B85 = 269.32151495544974
A86 = 1.4660765716752369
B86 = 366.0925238710978
A87 = 1.4835298641951802
B87 = 526.5843825754397
A88 = 1.5009831567151235
B88 = 822.0362215434371
A89 = 1.5184364492350666
B89 = 1460.3591091830504
A90 = 1.53588974175501
B90 = 3284.140008333149
A91 = 1.5533430342749532
B91 = 13132.558814615699
A92 = NaN
B92 = NaN
A93 = 1.5882496193148399
B93 = 13132.558814615746
A94 = 1.6057029118347832
B94 = 3284.140008333154
A95 = 1.6231562043547265
B95 = 1460.3591091830763
A96 = 1.6406094968746698
B96 = 822.0362215434386
A97 = 1.6580627893946132
B97 = 526.5843825754398
A98 = 1.6755160819145565
B98 = 366.0925238710944
A99 = 1.6929693744344996
B99 = 269.32151495545105
A100 = 1.710422666954443
B100 = 206.5139451448889
A101 = 1.7278759594743862
B101 = 163.45383275624584
A102 = 1.7453292519943295
B102 = 132.65374991010606
A103 = 1.7627825445142729
B103 = 109.86574409294496
A104 = 1.7802358370342162
B104 = 92.53417786804505
A105 = 1.7976891295541595
B105 = 79.04673299802278
A106 = 1.8151424220741028
B106 = 68.34545478725863
A107 = 1.8325957145940461
B107 = 59.7128129211021
A108 = 1.8500490071139892
B108 = 52.64823801244084
A109 = 1.8675022996339325
B109 = 46.793907407378285
A110 = 1.8849555921538759
B110 = 41.888543819998375
A111 = 1.9024088846738192
B111 = 37.737763288042345
A112 = 1.9198621771937625
B112 = 34.19452868165209
A113 = 1.9373154697137058
B113 = 31.145956139948833
A114 = 1.9547687622336491
B114 = 28.50421972781678
A115 = 1.9722220547535925
B115 = 26.200161478229536
A116 = 1.9896753472735358
B116 = 24.17872475892025
A117 = 2.007128639793479
B117 = 22.395639728453574
A118 = 2.0245819323134224
B118 = 20.814983371179334
A119 = 2.0420352248333655
B119 = 19.40735998527675
A120 = 2.059488517353309
B120 = 18.14852814981645
A121 = 2.076941809873252
B121 = 17.018353213199376
A122 = 2.0943951023931953
B122 = 16.00000000000001
A123 = 2.111848394913139
B123 = 15.079304781491635
A124 = 2.129301687433082
B124 = 14.244282419364172
A125 = 2.1467549799530254
B125 = 13.484736427134758
A126 = 2.1642082724729685
B126 = 12.791948101429744
A127 = 2.181661564992912
B127 = 12.158426916645855
A128 = 2.199114857512855
B128 = 11.577708763999679
A129 = 2.2165681500327987
B129 = 11.044191834358152
A130 = 2.234021442552742
B130 = 10.553002327284313
A131 = 2.251474735072685
B131 = 10.099883948873432
A132 = 2.2689280275926285
B132 = 9.681106501844804
A133 = 2.2863813201125716
B133 = 9.293389889328836
A134 = 2.303834612632515
B134 = 8.933840632380882
A135 = 2.321287905152458
B135 = 8.599898600960023
A136 = 2.3387411976724017
B136 = 8.28929212311677
A137 = 2.356194490192345
B137 = 8.000000000000009
A138 = 2.3736477827122884
B138 = 7.730219239153538
A139 = 2.3911010752322315
B139 = 7.478337543497337
A140 = 2.4085543677521746
B140 = 7.242909771951638
A141 = 2.426007660272118
B141 = 7.022637730172685
A142 = 2.443460952792061
B142 = 6.816352764167384
A143 = 2.4609142453120048
B143 = 6.623000721670309
A144 = 2.478367537831948
B144 = 6.441628920743991
A145 = 2.4958208303518914
B145 = 6.2713748257052995
A146 = 2.5132741228718345
B146 = 6.111456180001695
A147 = 2.530727415391778
B147 = 5.961162386262825
A148 = 2.548180707911721
B148 = 5.819846957171884
A149 = 2.5656340004316642
B149 = 5.68692088841033
A150 = 2.5830872929516078
B150 = 5.561846827822348
A151 = 2.600540585471551
B151 = 5.444133933992731
A152 = 2.6179938779914944
B152 = 5.333333333333325
A153 = 2.6354471705114375
B153 = 5.229034098089891
A154 = 2.652900463031381
B154 = 5.130859678871101
A155 = 2.670353755551324
B155 = 5.038464734729993
A156 = 2.6878070480712677
B156 = 4.951532311796629
A157 = 2.705260340591211
B157 = 4.869771328216012
A158 = 2.722713633111154
B158 = 4.792914328886284
A159 = 2.7401669256310974
B159 = 4.720715478384702
A160 = 2.7576202181510405
B160 = 4.6529487646503975
A161 = 2.775073510670984
B161 = 4.589406389574692
A162 = 2.792526803190927
B162 = 4.529897325727179
A163 = 2.8099800957108707
B163 = 4.474246021095028
A164 = 2.827433388230814
B164 = 4.422291236000327
A165 = 2.8448866807507573
B165 = 4.373884998340706
A166 = 2.8623399732707004
B166 = 4.328891665015867
A167 = 2.8797932657906435
B167 = 4.287187078897973
A168 = 2.897246558310587
B168 = 4.248657812007088
A169 = 2.91469985083053
B169 = 4.213200486694359
A170 = 2.9321531433504737
B170 = 4.180721167635926
A171 = 2.949606435870417
B171 = 4.151134818321896
A172 = 2.9670597283903604
B172 = 4.124364816503045
A173 = 2.9845130209103035
B173 = 4.100342523747666
A174 = 3.001966313430247
B174 = 4.079006904875362
A175 = 3.01941960595019
B175 = 4.060304193584478
A176 = 3.036872898470133
B176 = 4.044187601083137
A177 = 3.0543261909900767
B177 = 4.030617064982212
A178 = 3.07177948351002
B178 = 4.01955903611432
A179 = 3.0892327760299634
B179 = 4.01098630131752
A180 = 3.1066860685499065
B180 = 4.0048778405675876
A181 = 3.12413936106985
B181 = 4.001218717166593
A182 = 3.141592653589793
B182 = 4
C = (-26.628837678316305, 27.422790254134686)
D = (-16.136527338590206, 27.524657344811637)
h = Segment(C, D)
J = Point(i)
j = OrthogonalLine(V, xAxis)
k = OrthogonalLine(V, yAxis)
K = Intersect(k, yAxis)
L = Intersect(j, xAxis)
l = Segment(V, K)
m = Segment(V, L)
texte2: "\alpha=" + (LaTeX[α]) + ""
texte3: "a=" + poly1 + ""
n = Segment(I, M)
β = Angle(B, I, M)
p = Segment(I, B)
H = Point(a)
N = Point(a)
O = Point(b)
P = Point(b)
u = Vector(O, P)
v = Vector(N, H)
Q = Midpoint(I, B)