Les \;trois\;carrés:1^{ère}\;S

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Conjecture\
Ainsi\;à \;l'aide\;des\;formules\;d'addition,\;on\; a \;vérifié \;que:\
cos\left(\alpha+\beta   \right) =cos\left( \gamma  \right)\;\;\;et\;\;\; sin\left(\alpha+\beta   \right)=sin\left( \gamma  \right)\
alors\;\;\;\alpha+\beta=\gamma  

Curseur\; de\; commande\
 \;\;\;du\; côté\; des\; carrés

O

I

A

M

B

J

\;\;\;\;\;\;\;\;\;\;Remarque\
\;\;\;\alpha,\;\beta\;et\;\gamma\;restent\;fixes\
\left(  lorsque\; on\; change\; le\;côté \right) 

c = 0.7 (slider from 0.5 to 1.2, step 0.1)
O = (0, 2)
J: (0, 2 + c)
A: ((2 * c), 2)
M: ((3 * c), 2)
B: ((3 * c), c + 2)
I': (c, c + 2)
A': ((2 * c), c + 2)
I: (c, 2)
poly1, o, i, i', j = Polygon(O, I, I', J)
poly2, i_1, a, a', i'_1 = Polygon(I, A, A', I')
poly3, a_1, m, b, a'_1 = Polygon(A, M, B, A')
α = Angle(M, O, B)
β = Angle(M, I, B)
γ = Angle(M, A, B)
c_1 = Segment(O, B)
d = Segment(I, B)
e = Segment(A, B)
f = Segment(O, M)
s: α + β
texte2: "\alpha+\beta=" + (LaTeX[s]) + ""
c_α: cos(α)
c_β: cos(β)
c_s: cos(s)
c_γ: cos(γ)
s_α: sin(α)
s_β: sin(β)
texte3: "cos\left( \alpha \right) =" + (LaTeX[c_α]) + "\\
cos\left(\beta   \right)=" + (LaTeX[c_β]) + "\\
cos\left(   \gamma\right)  =" + (LaTeX[c_γ]) + ""
texte6: "cos\left( \alpha+\beta  \right) =cos\left(\alpha   \right) ×cos\left(\beta   \right)-sin\left( \alpha  \right)×sin\left(  \beta \right)\\ 
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=" + (LaTeX[c_α]) + "×" + (LaTeX[c_β]) + "-" + (LaTeX[s_α]) + "×" + (LaTeX[s_β]) + "=" + (LaTeX[c_γ]) + "  \\
\;\\
cos\left( \gamma  \right) =" + (LaTeX[c_γ]) + ""
s_γ: sin(γ)
O_1 = (0.7547227686127768, 8.720240781202698)
J_1: (0, 2 + c)
A_1: ((2 * c), 2)
M_1: ((3 * c), 2)
B_1: ((3 * c), c + 2)
I'_1: (c, c + 2)
A'_1: ((2 * c), c + 2)
I_1: (c, 2)
α_1 = Angle(M_1, O_1, B_1)
β_1 = Angle(M_1, I_1, B_1)
γ_1 = Angle(M_1, A_1, B_1)
c_5 = Segment(O_1, B_1)
d_1 = Segment(I_1, B_1)
e_1 = Segment(A_1, B_1)
f_1 = Segment(O_1, M_1)
s_2: α_1 + β_1
c_2: cos(α_1)
c_3: cos(β_1)
c_4: cos(γ_1)
s_4: sin(α_1)
s_3: sin(β_1)
texte3_1: "cos\left( \alpha \right) =" + (LaTeX[c_2]) + "\\
cos\left(\beta   \right)=" + (LaTeX[c_3]) + "\\
cos\left(   \gamma\right)  =" + (LaTeX[c_4]) + ""
s_1: sin(γ_1)
c_6 = 1 (slider from 1 to 5, step 0.1)
O_2 = (-1.9154366084954444, 6.802822020335419)
J_2: (0, 2 + c_6)
A_2: ((2 * c_6), 2)
M_2: ((3 * c_6), 2)
B_2: ((3 * c_6), c_6 + 2)
I'_2: (c_6, c_6 + 2)
A'_2: ((2 * c_6), c_6 + 2)
I_2: (c_6, 2)
α_2 = Angle(M_2, O_2, B_2)
β_2 = Angle(M_2, I_2, B_2)
γ_2 = Angle(M_2, A_2, B_2)
c_{10} = Segment(O_2, B_2)
d_2 = Segment(I_2, B_2)
e_2 = Segment(A_2, B_2)
f_2 = Segment(O_2, M_2)
s_6: α_2 + β_2
c_7: cos(α_2)
c_8: cos(β_2)
c_9: cos(γ_2)
s_8: sin(α_2)
s_7: sin(β_2)
texte3_2: "cos\left( \alpha \right) =" + (LaTeX[c_7]) + "\\
cos\left(\beta   \right)=" + (LaTeX[c_8]) + "\\
cos\left(   \gamma\right)  =" + (LaTeX[c_9]) + ""
s_5: sin(γ_2)
O_3 = (1.1842094243249357, 8.601306015005484)
J_3: (0, 2 + c_6)
A_3: ((2 * c_6), 2)
M_3: ((3 * c_6), 2)
B_3: ((3 * c_6), c_6 + 2)
I'_3: (c_6, c_6 + 2)
A'_3: ((2 * c_6), c_6 + 2)
I_3: (c_6, 2)
α_3 = Angle(M_3, O_3, B_3)
β_3 = Angle(M_3, I_3, B_3)
γ_3 = Angle(M_3, A_3, B_3)
c_{14} = Segment(O_3, B_3)
d_3 = Segment(I_3, B_3)
e_3 = Segment(A_3, B_3)
f_3 = Segment(O_3, M_3)
s_{10}: α_3 + β_3
c_{11}: cos(α_3)
c_{12}: cos(β_3)
c_{13}: cos(γ_3)
s_{12}: sin(α_3)
s_{11}: sin(β_3)
s_9: sin(γ_3)
J_4: (0, 2 + c_6)
A_5: ((2 * c_6), 2)
M_5: ((3 * c_6), 2)
B_5: ((3 * c_6), c_6 + 2)
I'_4: (c_6, c_6 + 2)
A'_4: ((2 * c_6), c_6 + 2)
I_5: (c_6, 2)
O_4 = (0.1659217999557871, 3.9372412219001465)
J_5: (0, 2 + c_6)
A_4: ((2 * c_6), 2)
M_4: ((3 * c_6), 2)
B_4: ((3 * c_6), c_6 + 2)
I'_5: (c_6, c_6 + 2)
A'_5: ((2 * c_6), c_6 + 2)
I_4: (c_6, 2)
α_4 = Angle(M_4, O_4, B_4)
β_4 = Angle(M_4, I_4, B_4)
γ_4 = Angle(M_4, A_4, B_4)
c_{18} = Segment(O_4, B_4)
d_4 = Segment(I_4, B_4)
e_4 = Segment(A_4, B_4)
f_4 = Segment(O_4, M_4)
s_{14}: α_4 + β_4
c_{15}: cos(α_4)
c_{16}: cos(β_4)
c_{17}: cos(γ_4)
s_{16}: sin(α_4)
s_{15}: sin(β_4)
s_{13}: sin(γ_4)
texte3_3: "sin\left( \alpha \right) =" + (LaTeX[s_α]) + "\\
cos\left(\beta   \right)=" + (LaTeX[s_β]) + "\\
cos\left(   \gamma\right)  =" + (LaTeX[s_γ]) + ""
O_5 = (6.615563747331009, 7.200518768682753)
J_6: (0, 2 + c)
A_6: ((2 * c), 2)
M_6: ((3 * c), 2)
B_6: ((3 * c), c + 2)
I'_6: (c, c + 2)
A'_6: ((2 * c), c + 2)
I_6: (c, 2)
α_5 = Angle(M_6, O_5, B_6)
β_5 = Angle(M_6, I_6, B_6)
γ_5 = Angle(M_6, A_6, B_6)
c_{22} = Segment(O_5, B_6)
d_5 = Segment(I_6, B_6)
e_5 = Segment(A_6, B_6)
f_5 = Segment(O_5, M_6)
s_{20}: α_5 + β_5
c_{19}: cos(α_5)
c_{20}: cos(β_5)
c_{21}: cos(γ_5)
s_{17}: sin(α_5)
s_{18}: sin(β_5)
s_{19}: sin(γ_5)
J_7: (0, 2 + c)
A_7: ((2 * c), 2)
M_7: ((3 * c), 2)
B_7: ((3 * c), c + 2)
I'_7: (c, c + 2)
A'_7: ((2 * c), c + 2)
I_7: (c, 2)
J_8: (0, 2 + c)
A_9: ((2 * c), 2)
M_9: ((3 * c), 2)
B_9: ((3 * c), c + 2)
I'_8: (c, c + 2)
A'_8: ((2 * c), c + 2)
I_9: (c, 2)
J_9: (0, 2 + c)
A_{10}: ((2 * c), 2)
M_{10}: ((3 * c), 2)
B_{10}: ((3 * c), c + 2)
I'_9: (c, c + 2)
A'_9: ((2 * c), c + 2)
I_{10}: (c, 2)
O_6 = (2.2737101564508455, 3.838128916735802)
J_{10}: (0, 2 + c)
A_8: ((2 * c), 2)
M_8: ((3 * c), 2)
B_8: ((3 * c), c + 2)
I'_{10}: (c, c + 2)
A'_{10}: ((2 * c), c + 2)
I_8: (c, 2)
α_6 = Angle(M_8, O_6, B_8)
β_6 = Angle(M_8, I_8, B_8)
γ_6 = Angle(M_8, A_8, B_8)
c_{26} = Segment(O_6, B_8)
d_6 = Segment(I_8, B_8)
e_6 = Segment(A_8, B_8)
f_6 = Segment(O_6, M_8)
s_{24}: α_6 + β_6
c_{23}: cos(α_6)
c_{24}: cos(β_6)
c_{25}: cos(γ_6)
s_{21}: sin(α_6)
s_{22}: sin(β_6)
s_{23}: sin(γ_6)
J_{11}: (0, 2 + c)
A_{11}: ((2 * c), 2)
M_{11}: ((3 * c), 2)
B_{11}: ((3 * c), c + 2)
I'_{11}: (c, c + 2)
A'_{11}: ((2 * c), c + 2)
I_{11}: (c, 2)
texte6_1: "sin\left( \alpha+\beta  \right) =sin\left(\alpha   \right) ×cos\left(\beta   \right)+cos\left( \alpha  \right)×sin\left(  \beta \right)\\ 
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=" + (LaTeX[s_α]) + "×" + (LaTeX[c_β]) + "+" + (LaTeX[c_α]) + "×" + (LaTeX[s_β]) + "=" + (LaTeX[s_γ]) + " \\
\;\\
sin\left( \gamma  \right) =" + (LaTeX[s_γ]) + ""
texte6_2: "cos\left( \alpha+\beta  \right) =cos\left(\alpha   \right) ×cos\left(\beta   \right)-sin\left( \alpha  \right)×sin\left(  \beta \right)\\ 
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=" + (LaTeX[c_α]) + "×" + (LaTeX[c_β]) + "-" + (LaTeX[s_α]) + "×" + (LaTeX[s_β]) + "=" + (LaTeX[c_{25}]) + "  \\
\;\\
cos\left( \gamma  \right) =" + (LaTeX[c_{25}]) + ""
texte3_4: "cos\left( \alpha \right) =" + (LaTeX[c_α]) + "\\
cos\left(\beta   \right)=" + (LaTeX[c_{16}]) + "\\
cos\left(   \gamma\right)  =" + (LaTeX[c_{17}]) + ""