Carr\303\251s congruents d'Ehrmann
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En utilisant les formules de MathWorld
with(plots):
dist:=proc(A,B) local U:=expand(A-B): sqrt(U[1]^2+U[2]^2) end:
vperp:=proc(V) [-V[2],V[1]]: end:
valign:=proc(A,B,M) local U,V: U:=expand(B-A): V:=expand(M-A): U[2]*V[1]-U[1]*V[2]: end:#---- M align\303\251 avec AetB -----------
carre_inscrit:=proc(M,N,P) #----- construit un carr\303\251 inscrit avec un cot\303\251 port\303\251 par NP;
local R,S, t0:
R:=P-vperp(P-N); S:=N-vperp(P-N); t0:=solve( valign(N,P,M+t*(R-M)),t ):
[M+t0*(N-M),M+t0*(P-M),M+t0*(R-M),M+t0*(S-M),M+t0*(N-M)]: #--- je referme le carr\303\251
end:
#-----------------------------------------------------------------------
a,b,c:= 2,5,8: # quelconque # 0,4,8 : # rectangle #-1,5,8: # 4,4*sqrt(3),8: #triangle \303\251quilat\303\251ral # 4,3,8 : # isoc\303\250le #
A:=[a,b]: B:=[0,0]: C:=[c,0]:
c1:=dist(B,C): c2:=dist(C,A): c3:=dist(A,B): p:=c1+c2+c3:
Delta:=sqrt(p/(p-c1)*(p-c2)*(p-c3) ): #---aire du triangle
L:='L':
eq:= evalf(numer(simplify((c1^2)/(c1-L)+(c2^2)/(c2-L)+(c3^2)/(c3-L)-2*Delta/L))):
L:=min(solve(eq,L));
Tr1:=[ c1^2/(c1-L),c2^2/(c2-L),c3^2/(c3-L)]:
Tr0:=[A,B,C]: P:=evalf([add(Tr1[i]*Tr0[i,1],i=1..3),add(Tr1[i]*Tr0[i,2],i=1..3)]/add(Tr1[i],i=1..3)):
carres:=evalf([carre_inscrit(P,A,B), carre_inscrit(P,B,C), carre_inscrit(P,C,A)]):
display([ plot([A,B,C,A], color=black) ,plot({[P,A],[P,B],[P,C]},color=blue)
,plottools[disk](P,0.1,color=red),plot(carres,color=cyan,filled,transparency=0.6),plot(carres,color=red,linestyle=5)
, textplot({ [op(A+[0,0.4]),"A"],[op(B+[-0.4,0]),"B"],[op(C+[0.4,0]),"C"],[op(P+[0.3,0.3]),"P"]}, color=black)
],scaling=constrained,axes=none);
Trouver un point P tel que les carr\303\251s inscrits dans les LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5MTY7RicvJSVzaXplR1EjMjBGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEifkYnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMRjU=PAB, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2J1EnJiM5MTY7RicvJSVzaXplR1EjMjBGJy8lJ2l0YWxpY0dGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGLy1GLDYvRi5GSkZPRi9GMkY1RjdGOUY7Rj1GP0ZBRkRGLw==PAC, LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEoJkRlbHRhO0YnLyUlc2l6ZUdRIzIwRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== PBC soient \303\251gaux
# ----- P un point int\303\251rieur au triangle ABC ; AB[i] liste des sommet du carr\303\251 construit sur AB ( resp CA et BC)
restart:with(plots):
align:=proc(P,Q,R) Re(evalc(P-Q))*Im(evalc(P-R))- Re(evalc(P-R))*Im(evalc(P-Q)): end:
C2xy:=proc(z) [Re(z),Im(z)]: end:
carre_sur_seg:=proc(M,N,t,L)
map(evalc,simplify([M+t*(N-M),M+t*(N-M)+L*(N-M)/abs(N-M),M+t*(N-M)+L*(N-M)/abs(N-M)+I*L*(N-M)/abs(N-M),M+t*(N-M)+I*L*(N-M)/abs(N-M),M+t*(N-M)])):
end:
Ehrmann:=proc(a,b,c)
local A,B,C,BC,CA,AB,eqBC,eqCA,eqAB,resBC,resCA,resAB,res,solxy,S1,S2,SUBS,T,P,K,carres,PT:
A:=a+I*b: B:=0: C:=c: P:=x+I*y:
BC:=carre_sur_seg(B,C,t1,L1): CA:=carre_sur_seg(C,A,t2,L2): AB:=carre_sur_seg(A,B,t3,L3):
eqBC:=evalc({align(C,P,BC[3]),align(B,P,BC[4])}): resBC:=subs(op(solve(eqBC,{L1,t1}) ),[L1,t1]):
eqCA:=evalc({align(A,P,CA[3]),align(C,P,CA[4])}): resCA:=subs(op(solve(eqCA,{L2,t2}) ),[L2,t2]):
eqAB:=evalc({align(B,P,AB[3]),align(A,P,AB[4])}): resAB:=subs(op(solve(eqAB,{L3,t3}) ),[L3,t3]):
res:=simplify(solve({resBC[1]-resCA[1],resBC[1]-resAB[1],resCA[1]-resAB[1]},{x,y})):#---- egalit\303\251 des cot\303\251s des carr\303\251s
solxy:=map(u -> subs(op(u),[Re(x),Re(y)]),evalc([allvalues(res)])):
K:=1:
S1:=t1=resBC[2],t2=resCA[2],t3=resAB[2],L1=resBC[1],L2=resCA[1],L3=resAB[1]: S2:=x=solxy[K,1],y=solxy[K,2]:
SUBS:=proc(L) map(C2xy,subs(S2,subs(S1,L)) ): end:
T:=SUBS([A,B,C,A]): PT :=[SUBS([A,P]),SUBS([B,P]),SUBS([C,P])]: carres:=[SUBS(BC),SUBS(CA),SUBS(AB)]:
display([ plot(T, color=black) ,plot(PT,color=blue), plot(carres,color=red,linestyle=3),
seq(plottools[disk](T[i],0.1,color=cyan),i=1..3),
textplot({ [op(T[1]+[0,0.4]),"A"],[op(T[2]+[-0.4,0]),"B"],[op(T[3]+[0.4,0]),"C"],[op(SUBS(P)+[0.3,0.3]),"P"]}, color=black)
],scaling=constrained,axes=none);
end:
#5,6,0,0,8,0:
a,b,c:= 4,4*sqrt(3),8: #triangle \303\251quilat\303\251ral #0,4,8 : # rectangle # 5,6,8: # quelconque # -1,5,8: # 4,3,8 : # isoc\303\250le #
Ehrmann(a,b,c);
#Ehrmann( 4,4*sqrt(3),8); Ehrmann( ;
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;
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