Gouttière\; de\; longueur\; minimale

I

A

B

C

D

H

M

N

N\left( x, l) \;\;\;\;et\;\;\;\;N'\left(  \Theta,l \right)  \right) \
\;l =MA+MB+MH

N'

Gouttière\; de\; longueur\; minimale

Le \;curseur\;m\;commande\;l'animation

Conjecturez\;les\;valeurs\; respectives \
de\; x\; et \;de\; \Theta\;pur\;lesquelles\;l\;atteint\
\;\;\;\;\;\;\;\;\;\;sa\;valeur\;minimale

Conjecture\;d'après\
*\;les\; coordonnées\; de\; N\; et\; N':\
l=14.6603\;m\;et\;\theta=0.5256\;rad\;pour\;x=2.9\;m\
*\;le\;tableur,\;à\;la\;\;ligne\;30,\;on\;retrouve:\
l=14.6603\;m\;et\;\theta=0.5256\;rad\;pour\;x=2.9\;m

m = 3.4 (slider from 0 to 6, step 0.1)
C = (10, 0)
B = (10, 6)
A = (0, 6)
I = Midpoint(A, B)
p = Circle(I, m)
D = (0, 0)
poly1, a, b, c, d = Polygon(A, B, C, D)
e = LineBisector(c)
H = Intersect(e, c)
f = Segment(I, H)
M = Intersect(p, f)
g = Segment(A, M)
h = Segment(M, B)
i = Segment(M, H)
j = Segment(M, I)
E = Midpoint(I, M)
q = Distance(I, M)
l: Distance[M, A] + Distance[M, B] + Distance[M, H]
N: (q, l)
k: y = y(N)
F = Point(k)
Θ = Angle(M, A, I)
N': (Θ, l)
D_1 = (6.200688278801232, 5.294662580338753)
A_1 = (6.200688278801232, 11.294662580338787)
B_1 = (16.20068827880123, 11.294662580338787)
C_1 = (16.20068827880123, 5.294662580338753)
poly1_1, a_1, b_1, c_1, d_1 = Polygon(A_1, B_1, C_1, D_1)
e_1 = LineBisector(c_1)
I_1 = Midpoint(A_1, B_1)
H_1 = Intersect(e_1, c_1)
f_1 = Segment(I_1, H_1)
M_1 = Point(f_1)
g_1 = Segment(A_1, M_1)
h_1 = Segment(M_1, B_1)
i_1 = Segment(M_1, H_1)
j_1 = Segment(M_1, I_1)
E_1 = Midpoint(I_1, M_1)
q_1 = Distance(I_1, M_1)
texte14: "\Theta=" + (LaTeX[Θ]) + ""
texte8: "IM=m=x=" + (LaTeX[q]) + ""
texte14_1: "x=" + (LaTeX[m]) + ""
texte11: "y=y(N)=y\left(N'   \right)= l=" + (LaTeX[l]) + ""
liste1: {poly1, Θ}
liste2: {poly1, Θ}
A1 = Name(m)
B1 = Name(Θ)
C1 = Name(l)
A2 = 0.1
C2 = 15.901999800039974
B2 = 0.01999733397315419 rad
A3 = 0.2
C3 = 15.80799680255744
B3 = 0.039978687123290786 rad
A4 = 0.3
C4 = 15.717983829094544
B4 = 0.05992815512121015 rad
A5 = 0.4
C5 = 15.631948963187568
B5 = 0.07982998571223562 rad
A6 = 0.5
C6 = 15.54987562112089
B6 = 0.09966865249116204 rad
A7 = 0.6
C7 = 15.471742649611336
B7 = 0.11942892601833874 rad
A8 = 0.7
C8 = 15.397524449091472
B8 = 0.13909594148207086 rad
A9 = 0.8
C9 = 15.32719112093773
B9 = 0.15865526218640139 rad
A10 = 0.9
C10 = 15.26070863670443
B10 = 0.17809293823119762 rad
A11 = 1
C11 = 15.198039027185569
B11 = 0.19739555984988078 rad
A12 = 1.1
C12 = 15.139140588936163
B12 = 0.2165503049760892 rad
A13 = 1.2
C13 = 15.083968105745953
B13 = 0.2355449807208632 rad
A14 = 1.3
C14 = 15.032473082471592
B14 = 0.2543680585532657 rad
A15 = 1.4
C15 = 14.984603988597737
B15 = 0.27300870308671055 rad
A16 = 1.5
C16 = 14.94030650891055
B16 = 0.2914567944778671 rad
A17 = 1.6
C17 = 14.89952379872535
B17 = 0.30970294454245645 rad
A18 = 1.7
C18 = 14.862196741208717
B18 = 0.3277385067805555 rad
A19 = 1.8
C19 = 14.828264204469138
B19 = 0.34555558058171226 rad
A20 = 1.9
C20 = 14.797663296253065
B20 = 0.3631470099461763 rad
A21 = 2
C21 = 14.770329614269007
B21 = 0.3805063771123649 rad
A22 = 2.1
C22 = 14.746197490364999
B22 = 0.3976279915221292 rad
A23 = 2.2
C23 = 14.725200226998131
B23 = 0.4145068745847862 rad
A24 = 2.3
C24 = 14.70727032465361
B24 = 0.4311387407187823 rad
A25 = 2.4
C25 = 14.692339699089636
B25 = 0.4475199751571698 rad
A26 = 2.5
C26 = 14.680339887498949
B26 = 0.4636476090008061 rad
A27 = 2.6
C27 = 14.671202242884295
B27 = 0.47951929199259624 rad
A28 = 2.7
C28 = 14.664858116140298
B28 = 0.49513326346840414 rad
A29 = 2.8
C29 = 14.661239025515522
B29 = 0.5104883219167758 rad
A30 = 2.9
C30 = 14.660276813294741
B30 = 0.52558379355161 rad
A31 = 3
C31 = 14.661903789690601
B31 = 0.5404195002705842 rad
A32 = 3.1
C32 = 14.666052864066183
B32 = 0.5549957273385867 rad
A33 = 3.2
C33 = 14.672657663724664
B33 = 0.5693131911006619 rad
A34 = 3.3
C34 = 14.681652640600127
B34 = 0.583373006993856 rad
A35 = 3.4
C35 = 14.69297316626478
B35 = 0.5971766580926776 rad
A36 = 3.5
C36 = 14.706555615733702
B36 = 0.6107259643892086 rad
A37 = 3.6
C37 = 14.72233744059949
B37 = 0.6240230529767569 rad
A38 = 3.7
C38 = 14.74025723206719
B38 = 0.6370703292756835 rad
A39 = 3.8
C39 = 14.760254774486064
B39 = 0.6498704494119476 rad
A40 = 3.9
C40 = 14.782271089990152
B40 = 0.6624262938331512 rad
A41 = 4
C41 = 14.806248474865697
B41 = 0.6747409422235527 rad
A42 = 4.1
C42 = 14.832130528261768
B42 = 0.6868176497586453 rad
A43 = 4.2
C43 = 14.859862173851607
B43 = 0.6986598247214632 rad
A44 = 4.3
C44 = 14.889389675038037
B44 = 0.7102710074866861 rad
A45 = 4.4
C45 = 14.920660644277369
B45 = 0.7216548508647612 rad
A46 = 4.5
C46 = 14.95362404707371
B46 = 0.7328151017865066 rad
A47 = 4.6
C47 = 14.988230201170422
B47 = 0.7437555842988599 rad
A48 = 4.7
C48 = 15.024430771438208
B48 = 0.7544801838344056 rad
A49 = 4.8
C49 = 15.062178760930767
B49 = 0.7649928327109102 rad
A50 = 4.9
C50 = 15.10142849854971
B50 = 0.7752974968121263 rad
A51 = 5
C51 = 15.142135623730951
B51 = 0.7853981633974483 rad
A52 = 5.1
C52 = 15.184257068535276
B52 = 0.7952988299854369 rad
A53 = 5.2
C53 = 15.227751037497146
B53 = 0.805003494254653 rad
A54 = 5.3
C54 = 15.272576985557496
B54 = 0.8145161449044872 rad
A55 = 5.4
C55 = 15.31869559437928
B55 = 0.8238407534186363 rad
A56 = 5.5
C56 = 15.366068747318506
B56 = 0.8329812666744317 rad
A57 = 5.6
C57 = 15.414659503298768
B57 = 0.8419416003422657 rad
A58 = 5.7
C58 = 15.464432069813892
B58 = 0.8507256330207998 rad
A59 = 5.8
C59 = 15.515351775261317
B59 = 0.8593372010553886 rad
A60 = 5.9
C60 = 15.567385040788245
B60 = 0.8677800939891389 rad
A61 = 6
C61 = 15.620499351813308
B61 = 0.8760580505981934 rad