DERIVE for Windows version 5.00 DfW file saved on 26 Aug 2000
:   ALTURAATRIANGULO(m, n, p, q, r, s):=n·(q - s) + m·(p - r) + y·(s - q) + x·(r - p) = 0
ALTURABTRIANGULO(m, n, p, q, r, s):=ALTURAATRIANGULO(p, q, r, s, m, n)
ALTURACTRIANGULO(m, n, p, q, r, s):=ALTURAATRIANGULO(r, s, m, n, p, q)
AREATRIANGULOV(a, b, c):=1/2·ABS(DET([a™1, a™2, 1; b™1, b™2, 1; c™1, c™2, 1]))
BARICENTROTRIANGULO(m, n, p, q, r, s):=[(r + p + m)/3, (s + q + n)/3]
CIRCUNCENTROTRIANGULO(m, n, p, q, r, s):=[1/2·(q·(- s^2 - r^2 + q·s) + p^2·s - n·(- s^2 - r^2 + q^2 + p^2) + n^2·(q - s) + m^2·(q - s))/(- q·r + p·s + n·(r - p) + m·(q - s)), - 1/2·(q^2·r - p·(r^2 + s^2) + p^2·r + n^2·(p - r) - m·(- s^2 - r^2 + q^2 + p^2) + m^2·(p - r))/(- q·r + p·s + n·(r - p) + m·(q - s))]
DISTANCIA(a, b, m, n, p):=ABS(p + n·b + m·a)·(m^2 + n^2)^(- 1/2)
DISTANCIAV(v, m, n, p):=ABS(p + n·v™2 + m·v™1)·(m^2 + n^2)^(- 1/2)
HOMOTETICODEVRESPECTOPRAZONK(u, v, p, q, k):=[p, q] + k·[u - p, v - q]
HOMOTETICODEVRESPECTOPRAZONKV(v, p, k):=p + k·(v - p)
INTERSECCIONRECTAS(©, ß, ¼, ´, ¿, µ):=[(ß·µ - ¼·¿)/(©·¿ - ß·´), (¼·´ - ©·µ)/(©·¿ - ß·´)]
INTERSECDIAGCUADR(m, n, p, q, r, s, t, u):=[(m·(p·(s - u) + t·(q - s)) - r·(q·t - p·u + n·(p - t)))/(- s·t + r·u - q·r + p·s + n·(t - p) + m·(q - u)), (s·(p·u - q·t) - n·(- r·u + q·(r - t) + p·u) + m·s·(q - u))/(- s·t + r·u - q·r + p·s + n·(t - p) + m·(q - u))]
INVERSODEVRESPECTOPPOTENCIAK(u, v, p, q, k):=[p, q] + (k·ABS([u - p, v - q])^(-2))·[u - p, v - q]
INVERSODEVRESPECTOPPOTENCIAKV(v, p, k):=p + (k·ABS(v - p)^(-2))·(v - p)
MACROALTURASTRIANGULO(m, n, p, q, r, s):=[[m, n; p, q; r, s; m, n], [ALTURAATRIANGULO(m, n, p, q, r, s), ALTURABTRIANGULO(m, n, p, q, r, s), ALTURACTRIANGULO(m, n, p, q, r, s)]]
MACROMEDIANASTRIANGULO(m, n, p, q, r, s):=[[m, n; p, q; r, s; m, n], [MEDIANAATRIANGULO(m, n, p, q, r, s), MEDIANABTRIANGULO(m, n, p, q, r, s), MEDIANACTRIANGULO(m, n, p, q, r, s)]]
MACROMEDIATRICESTRIANGULO(m, n, p, q, r, s):=[[m, n; p, q; r, s; m, n], [MEDIATRIZATRIANGULO(m, n, p, q, r, s), MEDIATRIZBTRIANGULO(m, n, p, q, r, s), MEDIATRIZCTRIANGULO(m, n, p, q, r, s)]]
MACRORECTAEULERTRIANGULO(m, n, p, q, r, s):=[[m, n; p, q; r, s; m, n], RECTAEULERTRIANGULO(m, n, p, q, r, s)]
MEDIANAATRIANGULO(m, n, p, q, r, s):=RECTAPORDOSPUNTOS(m, n, (p + r)/2, (q + s)/2)
MEDIANABTRIANGULO(m, n, p, q, r, s):=MEDIANAATRIANGULO(p, q, r, s, m, n)
MEDIANACTRIANGULO(m, n, p, q, r, s):=MEDIANAATRIANGULO(r, s, m, n, p, q)
MEDIATRIZ(a, b, c, d):=d^2 + c^2 - b^2 - a^2 + 2·y·(b - d) + 2·x·(a - c) = 0
MEDIATRIZATRIANGULO(m, n, p, q, r, s):=MEDIATRIZ(p, q, r, s)
MEDIATRIZBTRIANGULO(m, n, p, q, r, s):=MEDIATRIZATRIANGULO(p, q, m, n, r, s)
MEDIATRIZCTRIANGULO(m, n, p, q, r, s):=MEDIATRIZATRIANGULO(r, s, m, n, p, q)
MEDIATRIZV(a, b):=(b - a)™1·(x - ((a + b)/2)™1) + (b - a)™2·(y - ((a + b)/2)™2) = 0
NORMALUNITARIA(a, b, m, n, p):=IF(p + n·b + m·a > 0, - (1·(m^2 + n^2)^(- 1/2))·[m, n], (1·(m^2 + n^2)^(- 1/2))·[m, n])
NORMALUNITARIAV(v, m, n, p):=IF(p + n·v™2 + m·v™1 > 0, - (1·(m^2 + n^2)^(- 1/2))·[m, n], (1·(m^2 + n^2)^(- 1/2))·[m, n])
ORTOCENTROTRIANGULO(m, n, p, q, r, s):=[- (m·(r·s - p·q + n·(p - r)) + (q·s + p·r - n·(q + s) + n^2)·(q - s))/(- q·r + p·s + n·(r - p) + m·(q - s)), (- q·r·s + p·(q·s - r^2) + p^2·r + n·(r·s - p·q) + m·(r^2 - p^2 + n·(q - s)) + m^2·(p - r))/(- q·r + p·s + n·(r - p) + m·(q - s))]
PMEDIO(a, b, c, d):=[(a + c)/2, (b + d)/2]
PUNTOMEDIOV(a, b):=(a + b)/2
RECTACARTAPUNTOPENDIENTE(m, n, p):=IF(n ” 0, SOLVE(p + n·y + m·x = 0, y), SOLVE(p + n·y + m·x = 0, x))
RECTAEULERTRIANGULO(m, n, p, q, r, s):=[BARICENTROTRIANGULO(m, n, p, q, r, s), ORTOCENTROTRIANGULO(m, n, p, q, r, s), CIRCUNCENTROTRIANGULO(m, n, p, q, r, s)]
RECTAPARALELA(m, n, p, a, b):=m·(x - a) + n·(y - b) = 0
RECTAPARALELAV(m, n, p, v):=m·(x - v™1) + n·(y - v™2) = 0
RECTAPENDIENTEARECTACART(a, b, m):=IF(m = –, x = a, b - m·a - y + m·x = 0)
RECTAPENDIENTEARECTACARTV(a, m):=IF(m = –, x = a™1, a™2 - m·a™1 - y + m·x = 0)
RECTAPERPENDICULAR(m, n, p, a, b):=n·(x - a) - m·(y - b) = 0
RECTAPERPENDICULARV(m, n, p, v):=n·(x - v™1) - m·(y - v™2) = 0
RECTAPORDOSPUNTOS(a, b, c, d):=- b·c + a·d + y·(c - a) + x·(b - d) = 0
RECTAPORDOSPUNTOSV(u, v):=- u™2·v™1 + u™1·v™2 + y·(v™1 - u™1) + x·(u™2 - v™2) = 0
ROTACION(v, p, ang):=p + [(v™1 - p™1)·COS(ang) - (v™2 - p™2)·SIN(ang), (v™1 - p™1)·SIN(ang) + (v™2 - p™2)·COS(ang)]
SIMETRICO(a, b, m, n, p):=[a, b] + 2·DISTANCIA(a, b, m, n, p)·NORMALUNITARIA(a, b, m, n, p)
SIMETRICODEPUNTORESPECTOPUNTO(a, b, p, q):=2·[p, q] - [a, b]
SIMETRICODEPUNTORESPECTOPUNTOV(v, p):=2·p - v
SIMETRICOPUNTORESPECTORECTA(a, b, m, n, p):=[(- 2·p·m - 2·b·m·n + a·(n^2 - m^2))/(m^2 + n^2), (- 2·p·n - 2·a·m·n + b·(m^2 - n^2))/(m^2 + n^2)]
SIMETRICOPUNTORESPECTORECTAV(v, m, n, p):=[(- 2·p·m - 2·v™2·m·n + v™1·(n^2 - m^2))/(m^2 + n^2), (- 2·p·n - 2·v™1·m·n + v™2·(m^2 - n^2))/(m^2 + n^2)]
SIMETRICOV(v, m, n, p):=v + 2·DISTANCIAV(v, m, n, p)·NORMALUNITARIAV(v, m, n, p)
macro(a, b, c, d, e, f):=[[a, 0], [0, b], [c, 0], [0, d], [e, 0], [0, f], a·b - a·y + - b·x = 0, - d·e + e·y + d·x = 0, [a·e·(d - b)/(a·d - b·e), b·d·(a - e)/(a·d - b·e)], b·c - c·y + - b·x = 0, - e·f + e·y + f·x = 0, [c·e·(b - f)/(b·e - c·f), b·f·(e - c)/(b·e - c·f)], c·d - c·y + - d·x = 0, - a·f + a·y + f·x = 0, [a·c·(d - f)/(a·d - c·f), d·f·(a - c)/(a·d - c·f)], b·e·(a·(b·(c·(d - f) + e·f) - d·e·f) - c·d·e·(b - f))/((a·d - b·e)·(c·f - b·e)) + y·((- d·e + c·d + b·e)/d + c·f·(c - e)/(b·e - c·f) + b·e^2·(b - d)/(d·(a·d - b·e))) + x·(b·e·(b - d)/(a·d - b·e) + b·e·(b - f)/(b·e - c·f)) = 0]
ang:=
hCross:=APPROX(- 20833333333333331/100000000000000000)
vCross:=APPROX(4147727272727273/2500000000000000)
©:=
´:=
µ:=
¼:=
¿:=
ß:=
# ÿÿ   CTextObj        x    ÿ[{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fmodern\fcharset2 DfW5 Printer;}}
{\colortbl\red0\green0\blue0;}
\deflang1034\pard\plain\f2\fs24 Una demostraci\'f3n interesante del teorema de Papus con DERIVE.
\par Primero se introduce como UTILITY FILE el fichero adecuado para estudiar la colinealidad. 
\par Se toman los ejes coordenados para mayor facilidad de c\'e1lculo para DERIVE.
\par Sobre Ox se toman A,C,E y sobre Oy se toman B,D,F.
\par Lo que sigue demuestra que los puntos AB\'8dDE, BC\'8dEF y DE\'8dFA est\'e1n alineados.
\par }
ÿÿ   CExpnObj8   „         User      ð¿      LOAD("FuncionesBasicas.mth")€8   œ   h   ¨    User      ð¿      [a,0]€8   ´   h   À    User      ð¿      [0,b]€8   Ì   h   Ø    User      ð¿      [c,0]€8   ä   h   ð    User      ð¿      [0,d]€8   ü   h      User      ð¿      [e,0]€8     h       User      ð¿      [0,f]€8   ,  H  8   User      ð¿      RECTAPORDOSPUNTOSV([a,0],[0,b])€H  D  ð  P  Simp(#8)ü©ñÒMb`?	      -b*x-a*y+a*b=0€8   \  H  h   User      ð¿
      RECTAPORDOSPUNTOSV([0,d],[e,0])€P  t  è  €  
Simp(User){®Gázt?      d*x+e*y-d*e=0€8   Œ  ˜  ˜   User      ð¿      &INTERSECCIONRECTAS(-b,-a,a*b,d,e,-d*e)€   ¤    È  	Simp(#12)ü©ñÒMb?      )[a*e*(d-b)/(a*d-b*e),b*d*(a-e)/(a*d-b*e)]€8   Ô  H  à   User      ð¿      RECTAPORDOSPUNTOSV([c,0],[0,b])€H  ì  ð  ø  	Simp(#14)yé&1¬|?      -b*x-c*y+b*c=0€8     H     User      ð¿      RECTAPORDOSPUNTOSV([0,f],[e,0])€P    è  (  	Simp(#16){®Gázt?      f*x+e*y-e*f=0€8   4  ˜  @   User      ð¿      &INTERSECCIONRECTAS(-b,-c,b*c,f,e,-e*f)€   L    p  	Simp(#18)/Ý$•?      )[c*e*(b-f)/(b*e-c*f),b*f*(e-c)/(b*e-c*f)]€8   |  H  ˆ   User      ð¿      RECTAPORDOSPUNTOSV([c,0],[0,d])€H  ”  ð     	Simp(#20){®Gázt?      -d*x-c*y+c*d=0€8   ¬  H  ¸   User      ð¿      RECTAPORDOSPUNTOSV([0,f],[a,0])€P  Ä  è  Ð  	Simp(#22)¸…ëQ¸Ž?      f*x+a*y-a*f=0€8   Ü  ˜  è   User      ð¿      &INTERSECCIONRECTAS(-d,-c,c*d,f,a,-a*f)€   ô      	Simp(#24)9´Èv¾ŸŠ?      )[a*c*(d-f)/(a*d-c*f),d*f*(a-c)/(a*d-c*f)]€8   $     x   User      ð¿      AREATRIANGULOV([a*e*(d-b)/(a*d-b*e),b*d*(a-e)/(a*d-b*e)],[c*e*(b-f)/(b*e-c*f),b*f*(e-c)/(b*e-c*f)],[a*c*(d-f)/(a*d-c*f),d*f*(a-c)/(a*d-c*f)])€˜  „       	Simp(#26){®GázÄ?      0€   œ    ö   ÿ§{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fmodern\fcharset2 DfW5 Printer;}}
{\colortbl\red0\green0\blue0;}
\deflang1034\pard\plain\f2\fs24 De la misma forma se puede proceder para muchas conjeturas a las que se puede llegar mediante la experimentaci\'f3n.
\par 
\par Para ver gr\'e1ficamente de lo que se trata se puede hacer una gr\'e1fica de todo ello
\par }
€8     È  &   User      ð¿      gRECTAPORDOSPUNTOSV([a*e*(d-b)/(a*d-b*e),b*d*(a-e)/(a*d-b*e)],[c*e*(b-f)/(b*e-c*f),b*f*(e-c)/(b*e-c*f)])€8   2     ž  	Simp(#28)“V-Â?      ±x*(b*e*(b-d)/(a*d-b*e)+b*e*(b-f)/(b*e-c*f))+y*(b*e^2*(b-d)/(d*(a*d-b*e))+c*f*(c-e)/(b*e-c*f)+(b*e+c*d-d*e)/d)+b*e*(a*(b*(c*(d-f)+e*f)-d*e*f)-c*d*e*(b-f))/((a*d-b*e)*(c*f-b*e))=0€8   ª        User      ð¿      ÿÀmacro(a,b,c,d,e,f):=[[a,0],[0,b],[c,0],[0,d],[e,0],[0,f],-b*x-a*y+a*b=0,d*x+e*y-d*e=0,[a*e*(d-b)/(a*d-b*e),b*d*(a-e)/(a*d-b*e)],-b*x-c*y+b*c=0,f*x+e*y-e*f=0,[c*e*(b-f)/(b*e-c*f),b*f*(e-c)/(b*e-c*f)],-d*x-c*y+c*d=0,f*x+a*y-a*f=0,[a*c*(d-f)/(a*d-c*f),d*f*(a-c)/(a*d-c*f)],x*(b*e*(b-d)/(a*d-b*e)+b*e*(b-f)/(b*e-c*f))+y*(b*e^2*(b-d)/(d*(a*d-b*e))+c*f*(c-e)/(b*e-c*f)+(b*e+c*d-d*e)/d)+b*e*(a*(b*(c*(d-f)+e*f)-d*e*f)-c*d*e*(b-f))/((a*d-b*e)*(c*f-b*e))=0]€       #   Ï{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fmodern\fcharset2 DfW5 Printer;}}
{\colortbl\red0\green0\blue0;}
\deflang1034\pard\plain\f2\fs24 
\par }
€8   /  ø   ;   User      ð¿      macro(1,1,2,3,-2,4)ÿÿ   CDispOleObjå   G  %  w  ÿÿ  	 CDispItem                ˆ ÐÏà¡±á                >  þÿ	                               þÿÿÿ                    ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿýÿÿÿþÿÿÿþÿÿÿþÿÿÿýÿÿÿýÿÿÿýÿÿÿýÿÿÿ	   
                                                                      !   "   #   $   %   &   '   (   )   *   +   ,   -   .   /   0   1   2   3   4   5   6   7   8   9   :   ;   <   =   >   ?   @   A   B   C   D   E   F   G   H   I   J   K   L   M   N   O   P   Q   R   S   T   U   V   W   X   Y   Z   [   \   ]   ^   _   `   a   b   c   d   e   f   g   h   i   j   k   l   m   n   o   p   q   r   s   t   u   v   w   x   y   z   {   |   }   ~      €   R o o t   E n t r y                                               ÿÿÿÿÿÿÿÿ         À      F            €?Ó*XÀ   À        O l e                                                         
  ÿÿÿÿÿÿÿÿÿÿÿÿ                                               C O N T E N T S                                                   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       Nt      C o m p O b j                                                        ÿÿÿÿ                                       {       þÿÿÿ   þÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ                                                               þÿ
  ÿÿÿÿ      À      F5   Imagen (Mapa de bits independiente del dispositivo ) ÿÿÿÿ   
   StaticDib ô9²q                                                                                                                                                                                                                                                                                                                                                    ‚   ƒ   „   …   †   ‡   ˆ   ‰   Š   ‹   Œ      Ž         ‘   ’   “   ”   •   –   —   ˜   ™   š   ›   œ      ž   Ÿ       ¡   ¢   £   ¤   ¥   ¦   §   ¨   ©   ª   «   ¬   ­   ®   ¯   °   ±   ²   ³   ´   µ   ¶   ·   ¸   ¹   º   »   ¼   ½   ¾   ¿   À   Á   Â   Ã   Ä   Å   Æ   Ç   È   É   Ê   Ë   Ì   Í   Î   Ï   Ð   Ñ   Ò   Ó   Ô   Õ   Ö   ×   Ø   Ù   Ú   Û   Ü   Ý   Þ   ß   à   á   â   ã   ä   å   æ   ç   è   é   ê   ë   ì   í   î   ï   ð   ñ   ò   ó   ô   õ   ö   ÷   ø   ù   ú   û   ü   ý   þ   ÿ                      	  
                                               !  "  #  $  %  &  '  (  )  *  +  ,  -  .  /  0  1  2  3  4  5  6  7  8  9  :  ;  <  =  >  ?  @  A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z  [  \  ]  ^  _  `  a  b  c  d  e  f  g  h  i  j  k  l  m  n  o  p  q  r  s  t  u  v  w  x  y  z  {  |  }  ~    €    ‚  ƒ  „  …  †  ‡  ˆ  ‰  Š  ‹  Œ    Ž      ‘  ’  “  ”  •  –  —  ˜  ™  š  ›  œ    ž  Ÿ     ¡  ¢  £  ¤  ¥  ¦  §  ¨  ©  ª  «  ¬  ­  ®  ¯  °  ±  ²  ³  ´  µ  ¶  ·  ¸  ¹  º  »  ¼  ½  ¾  ¿  À  Á  Â  Ã  Ä  Å  Æ  Ç  È  É  Ê  Ë  Ì  Í  Î  Ï  Ð  Ñ  Ò  Ó  Ô  Õ  Ö  ×  Ø  Ù  Ú  Û  Ü  Ý  Þ  ß  à  á  â  ã  ä  å  æ  ç  è  é  ê  ë  ì  í  î  ï  ð  ñ  ò  ó  ô  õ  ö  ÷  ø  ù  ú  û  ü  ý  þ  ÿ                     	  
                                             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