Thank you very much for your task.
I try to find out a matrice 4X4 whuch calculates coordinates A->N, B->P, C->C and E->M.
it would take into account translation, rotation and ratio

will you take my example again and show that the procedure works. Thank you.

for i while (x := S || (i + 1))::And(algebraic, Not(name)) do
    S1 += x;
end do;
Error, recursive operator assignment
Sorry this instruction does not work.
 

@rlopez 
It is a very good prg.
Will you draw me a figure on an example ? Thank you.

@Kitonum 
For instance

A, B, C := <-3, 1, 2>, <-2, -1, 1>, <0, 3, -3>;
                 [&uminus0;3]  [&uminus0;2]  [    0     ]
                 [          ]  [          ]  [          ]
      A, B, C := [    1     ], [&uminus0;1], [    3     ]
                 [          ]  [          ]  [          ]
                 [    2     ]  [    1     ]  [&uminus0;3]

alpha := [a, b, c];
                       alpha := [a, b, c]

G := (A*alpha[1] + B*alpha[2] + C*alpha[3])/add(alpha);
G := Vector(3, {(1) = Typesetting:-mfrac(Typesetting:-mrow(Types\

  etting:-mo("&uminus0;"), Typesetting:-mrow(Typesetting:-mn("3"\

  ), Typesetting:-mo("&InvisibleTimes;"), Typesetting:-mi("a")),\

   Typesetting:-mo("&minus;"), Typesetting:-mrow(Typesetting:-mn\

  ("2"), Typesetting:-mo("&InvisibleTimes;"), Typesetting:-mi("b\

  "))), Typesetting:-mrow(Typesetting:-mi("a"), Typesetting:-mo(\

  "&plus;"), Typesetting:-mi("b"), Typesetting:-mo("&plus;"), 

   Typesetting:-mi("c"))), (2) = Typesetting:-mfrac(Typesetting:\

  -mrow(Typesetting:-mi("a"), Typesetting:-mo("&minus;"), 

   Typesetting:-mi("b"), Typesetting:-mo("&plus;"), Typesetting:\

  -mrow(Typesetting:-mn("3"), Typesetting:-mo("&InvisibleTimes;"\

  ), Typesetting:-mi("c"))), Typesetting:-mrow(Typesetting:-mi("\

  a"), Typesetting:-mo("&plus;"), Typesetting:-mi("b"), 

   Typesetting:-mo("&plus;"), Typesetting:-mi("c"))), (3) = 

   Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mrow(Typese\

  tting:-mn("2"), Typesetting:-mo("&InvisibleTimes;"), 

   Typesetting:-mi("a")), Typesetting:-mo("&plus;"), 

   Typesetting:-mi("b"), Typesetting:-mo("&minus;"), 

   Typesetting:-mrow(Typesetting:-mn("3"), Typesetting:-mo("&Inv\

  isibleTimes;"), Typesetting:-mi("c"))), Typesetting:-mrow(Type\

  setting:-mi("a"), Typesetting:-mo("&plus;"), Typesetting:-mi("\

  b"), Typesetting:-mo("&plus;"), Typesetting:-mi("c")))})

solve({G[1] = A, G[2] = B, G[3] = C}, {a, b, c});
Error, invalid input: solve expects its 1st argument, eqs, to be of type {`and`, `not`, `or`, algebraic, relation(algebraic), ({list, set})({`and`, `not`, `or`, algebraic, relation(algebraic)})}, but received {(-3*a-2*b)/(a+b+c) = (Vector(3, {(1) = -3, (2) = 1, (3) = 2})), (a-b+3*c)/(a+b+c) = (Vector(3, {(1) = -2, (2) = -1, (3) = 1})), (2*a+b-3*c)/(a+b+c) = (Vector(3, {(1) = 0, (2) = 3, (3) = -3}))}
Why this program fails; Thank you.

@tomleslie  Very good prg using geomety, Thang you very much;

@sand15 
Yes, weights may be negative

Tomlesie, I thank you yout program is very good.
I want now to draw equilateral hyperbole
  orthocenter(H, Tr1):
   evalf(coordinates(H));
   #conic(Hyp,[P1,P2,P3,P4,H],[x,y]);
But the computer crashes
Perhaps we should model the equation of this hyperbole?

Very good prg
line(L1,[P1,P3]):
line(L2,[P1,P4]):
ntersection(G,L1,L2);#gravity center of P1P2P3P4
G1:=convert(coordinates(G),Vector);
G:=<<-(5*sqrt(2))/2>|<(5*sqrt(2))/2>>;
pointplot(G,color = blue, symbol = diamond)
E - Euler center of P1,P2,P3,P4
I don(t know how yo place these points. 

 #p1 := polygonplot([P1,P2,P3,P4], color = blue, filled=true,transparency = 0.7):
   #p2 := polygonplot([o1,o2,o3,o4], color = red, filled=true,transparency = 0.8):
   display
   ( [ draw
       ( [ seq
           ( [ `||`(P, j)(color = black, symbol = solidcircle, symbolsize = 12),
               `||`(o, j)(color = black, symbol = solidcircle, symbolsize = 12),
               `||`(Tr, j)(color = green),
               `||`(Elc, j),
               `||`(seg, j)(color=black,filled=true)

             ][],
             j=1..4
           ),
           cir(color = blue)#p1,p2
When i take off #, I got massage of error with polygonplot. How to correct. Thank you.

I am sorry. I got still these messages
 

triangle: 0 < -P1+P2+P4
Error, (in geometry:-triangle) not enough information: the triangle is not defined
Error, (in geometry:-EulerCircle) wrong type of argument, expecting a triangle
triangle: 0 < -P1+P2+P3
Error, (in geometry:-triangle) not enough information: the triangle is not defined
Error, (in geometry:-EulerCircle) wrong type of argument, expecting a triangle
triangle: 0 < -P3+P2+P4
Error, (in geometry:-triangle) not enough information: the triangle is not defined
Error, (in geometry:-EulerCircle) wrong type of argument, expecting a triangle
triangle: 0 < -P1+P3+P4
Error, (in geometry:-triangle) not enough information: the triangle is not defined
Error, (in geometry:-EulerCircle) wrong type of argument, expecting a triangle
                              cir

                   ngon1 := [P1, P2, P3, P4]

Error, (in geometry:-coordinates) wrong type of argument
                   ngon2 := [P1, P2, P3, P4]

Error, (in geometry:-coordinates) wrong type of argument
Error, (in geometry:-coordinates) wrong type of argument
 

Sorry I need still your help. Could we still better use the computer to condense the reports ( 1,2,3,4) ?
 

  restart:
  with(geometry):
  with(plots):
  _EnvHorizontalName = 'x':
  _EnvVerticalName = 'y':
   R := 5:
   ang := [3/4*Pi, -(3*Pi)/4, -Pi/6,4*Pi/9]:
   seq
   ( point
     ( `||`(P, i),
       [ R*cos(ang[i]), R*sin(ang[i])]
     ),
     i = 1 .. 4
   ):
   seq
   ( dsegment
     ( `||`(seg, i),
       [ `||`(P, i),
         `||`(P, irem(i, 4) + 1)
       ]
     ),
     i = 1 .. 4
   ):
   triangle(Tr1,[P1,P2,P4]):
   EulerCircle(Elc1,Tr1,'centername'=o1):
   triangle(Tr2,[P1,P2,P3]):
   EulerCircle(Elc2,Tr2,'centername'=o2):
   triangle(Tr3,[P3,P2,P4]):
   EulerCircle(Elc3,Tr3,'centername'=o3):
   triangle(Tr4,[P1,P3,P4]):
   EulerCircle(Elc4,Tr4,'centername'=o4):
   circle(cir, [point(OO, [0, 0]), R]):
   ngon1 :=[P1,P2,P3,P4]:
   poly1:=polygonplot(ngon1,color=blue,filled=true):
   ngon2 :=[P1,P2,P3,P4]:
   poly2:=polygonplot(ngon2,color=yellow,filled=true):
   display
   ( [ draw
       ( [ P1(color = black, symbol = solidcircle, symbolsize = 12),
           P2(color = black, symbol = solidcircle, symbolsize = 12),
           P3(color = black, symbol = solidcircle, symbolsize = 12),
           P4(color = black, symbol = solidcircle, symbolsize = 12),
           o1(color = black, symbol = solidcircle, symbolsize = 12),
           o2(color = black, symbol = solidcircle, symbolsize = 12),
           o3(color = black, symbol = solidcircle, symbolsize = 12),
           o4(color = black, symbol = solidcircle, symbolsize = 12),
           seg1,
           seg2,
           seg3,
           seg4,
           Tr1(color=green),Tr2(color=green),Tr3(color=green),Tr4(color=green),
           Elc1,Elc2,Elc3,Elc4,
           cir(color = blue)
         ]
       ),
       textplot
       ( [ seq
           ( [ coordinates(`||`(P, i))[],
               convert(`||`(P, i), string)
             ],
             i=1..4
           )
         ],
         [ seq
           ( [ coordinates(`||`(o, i))[],
               convert(`||`(o, i), string)
             ],
             i=1..4
           )
         ],
            
         align=[above, right]
       )
     ],
     axes=none
   );

Error, (in plots:-polygonplot) incorrect arguments for creating polygons structure
Error, (in plots:-polygonplot) incorrect arguments for creating polygons structure
Error, (in plots:-textplot) unexpected option: [[(1/2)*((125/4)*2^(1/2)+((5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))^2*(-(5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))-((5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))*(-(5/4)*2^(1/2)+(5/2)*cos((4/9)*Pi))^2-((5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))*(-(5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))^2+(-(5/4)*2^(1/2)+(5/2)*cos((4/9)*Pi))^2*(-(5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi)))/(-25/4-(25/4)*2^(1/2)*cos((4/9)*Pi)), -(1/2)*((5/2)*2^(1/2)*((-(5/4)*2^(1/2)+(5/2)*cos((4/9)*Pi))^2+((5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))^2)-(5/2)*2^(1/2)*((-(5/4)*2^(1/2)+(5/2)*cos((4/9)*Pi))^2+(-(5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))^2)-(-(5/4)*2^(1/2) ... 2^(1/2)*sin((4/9)*Pi)+(25/16)*2^(1/2)*3^(1/2)-(25/16)*2^(1/2)), "o4"]]

Thank you very much for your help.
 

solve({x = 1 + 5*t, y = 1 + 3*t, 5*x + 3*y + 1 = 0}, {t, x, y});
Warning, solving for expressions other than names or functions is not recommended.
Thank you.

Thank you very much. How to leave fixed the circumscribed triangle P1P2P2 and make mobile the triangle inscribed AM2 M3 ? Then add the letters on figure M1, M2, A. Note that the M1M2 segment remains tangent to the inscribed circle.

ProjPL:=proc(P,A,B)
local H,x1,x2,x33,y1,y2,y3,t:
A:= <x1,y1>;
B:= <x2,y2>;
P :=<x3, y3>; 
H:=A+t*(B-A); solve( (P-H)^+ . (B-A), t );
'H':=eval(H, t=`%` );
end:
Error, (in ProjPL) invalid left hand side of assignment/ How to correct this error ? . Thank you.