Triangle\ rectangle\ et\ cercle : j'apprends\  le\ cours\ !

on\ sait\ que\ le\ triangle\ est\ rectangle

\textbf{Si}\ un\ triangle\ est\ rectangle,\

Si\ l’un\ des\ côtés\ d’un\ triangle\ est\ un\ diamètre\ 
d’un\ cercle\ et\ son\ troisième\ sommet\ est\ situé\ sur\ ce\ cercle,

Si\ la\ longueur\ de\ la\ médiane\ issue\ d’un\ 
sommet\ d’un\ triangle\ est\ égale\ à\ la\ moitié\ 
de\ la\ longueur\ du\ côté\ opposé,

\textbf{alors}\ la\ longueur\ de\ la\ médiane\ issue\ du\ sommet\ de\ l’angle\ droit\
est\ égale\ à\ la\ moitié\ de\ la\ longueur\
de \ l’hypoténuse .

alors\ ce\ triangle\ est\ rectangle\ en\ ce\ troisième\ sommet.

alors\ ce\ triangle\ est\ rectangle\ en\ ce\ sommet.

on\ cherche\ à\ démontrer\ que\ le\ triangle\ est\ rectangle

on\ sait\ au\ départ\ que

on\ en\ déduit\ que:

on\ utilise :

B

A

C

I

\textcolor{blue}{AI}=\frac{1}{2}\times \textcolor{orange}{BC}

"si..."

on\ sait\ au\ départ\ que

on\ en\ déduit\ que :

E

F

G

EFG\ est\ rectangle\ en\ G

on\ en\ déduit\ que:

K

L

M

I

KLM\ est\ rectangle\ en\ M

on\ sait\ au\ départ:

on\ utilise :

on\ utilise :

MathsEnClair.com

A_1 = (-3.8056116960214665, 2.003032659370451)
D = (-2.821907722542769, 3.3650843149563445)
a = Segment(A_1, D)
b = OrthogonalLine(D, a)
E = Point(b)
c = Segment(A_1, E)
d = Segment(D, E)
α = Angle(A_1, D, E)
B = (-4.7640924906930175, -5.563920982773403)
A = (-3.8560580536357585, -4.3784315788375325)
e = Segment(B, A)
f = OrthogonalLine(A, e)
C = Point(f)
g = Segment(B, C)
h = Segment(C, A)
β = Angle(B, A, C)
I = Midpoint(B, C)
i = Segment(A, I)
j = true
k = true
l = true
F = (9.007763138008743, 2.230041268634767)
G = (9.890574396258856, 3.1632988844991754)
p = Circle(F, G)
H = Point(p)
m = Line(H, F)
J = Intersect(p, m, 2)
K = Point(p)
n = Segment(H, J)
q = Segment(F, H)
r = Segment(F, J)
poly1, j_1, h_1, k_1 = Polygon(H, K, J)
o = true
s = true
L = (6.235218994975025, -5.554747922495139)
u = Vector(F, L)
H' = Translate(H, u)
K' = Translate(K, u)
J' = Translate(J, u)
poly1', j', h', k' = Polygon(H', K', J')
p' = Translate(p, u)
F' = Translate(F, u)
F'_1 = Translate(F, u)
H'_1 = Translate(H, u)
q' = Segment(F'_1, H'_1)
F'_2 = Translate(F, u)
J'_1 = Translate(J, u)
r' = Segment(F'_2, J'_1)
γ = Angle(H'_1, K', J'_1)
t = true
M = (20.438412113657705, 3.403803214200526)
N = (24.077039326041078, 1.948352329247169)
a_1 = Segment(M, N)
b_1 = OrthogonalLine(M, a_1)
O = Point(b_1)
c_1 = Segment(N, O)
P = Midpoint(c_1)
d_1 = Segment(P, M)
e_1 = Segment(P, N)
f_1 = Segment(P, O)
g_1 = Segment(M, O)
Q = (19.23390103645493, -5.830781711020776)
v = Vector(P, Q)
P' = Translate(P, v)
M' = Translate(M, v)
i_1 = Segment(P', M')
M'_1 = Translate(M, v)
O' = Translate(O, v)
l_1 = Segment(M'_1, O')
M'_2 = Translate(M, v)
N' = Translate(N, v)
m_1 = Segment(M'_2, N')
P'_1 = Translate(P, v)
N'_1 = Translate(N, v)
n_1 = Segment(P'_1, N'_1)
P'_2 = Translate(P, v)
O'_1 = Translate(O, v)
p_1 = Segment(P'_2, O'_1)
δ = Angle(O', M', N')
w = true
o_1 = true
q_1 = true