BISSEC := proc (P, U, V) local a, b, eq1, eq2, M1, M2, t, PU, PV, bissec1, bissec2; a := (P-U)/Norm(P-U, 2)+(P-V)/Norm(P-V, 2); M1 := P+a*t; b := (P-U)/Norm(P-U, 2)-(P-V)/Norm(P-V, 2); M2 := P+b*t; eq1 := op(eliminate({x = M1[1], y = M1[2]}, t)); eq2 := op(eliminate({x = M2[1], y = M2[2]}, t)); P := convert(P, list); U := convert(U, list); V := convert(V, list); PU := plot([P, U]); PV := plot([P, V]); bissec1 := implicitplot(op(eq1[2]), x = 0 .. 5, y = 0 .. 10, color = red); bissec2 := implicitplot(op(eq2[2]), x = 0 .. 5, y = 0 .. 10, color = green); display([bissec1, bissec2, PU, PV], scaling = constrained) end proc;

A := `<,>`(4, 8); B := `<,>`(4, 2); C := `<,>`(1, 4);
debug(BISSEC);
                             BISSEC
BISSEC(A, B, C);

P := [3/2, -42/25]; Q := [22649/5523, -2304/1841]; lma := 1.627112258;

with(geometry); point(Pp, P[1], P[2]); point(Qp, Q[1], Q[2]);
ellipse(ell, ['foci' = [Pp, Qp], 'MajorAxis' = lma]);
Error, (in geometry:-ellipse) the given polynomial/equation is not an algebraic representation of a ellipse

Why this error? Thank you

How to use LinearAlgebra to find characteristics of ellipse ?

 H2:=[4.4854,3.378]:H5:=[6.5922,2.0388]:

car_2som_opp(H2,H5) don't work with this message : Error invalid subscript selector. Thank you.

your code geo2 works in a fichier but not in the other.
How to correct hhese errors ? Thank you.
restart;
#Construction de 3 cercles tangents entre-eux et aux côtés d'un triangle
#a, b et c sont les 3 côtés, A,B et C les 3 sommets Ag, Bg, Cg les 3 angles, r1, r2 r3 les rayons des cercles

 

#A1A2=x, B1B2=y, C1C2=z, x²=4r1*r2, y²=4*r1r3, z²=4r1r2, r1=yz/2xn r2=xz/2y, r3=xy/2z

with(plots);
;
A := [-1, 5]; B := [-7, -1]; C := [12, -1];
a := sqrt((B[1]-C[1])^2+(B[2]-C[2])^2); b := sqrt((A[1]-C[1])^2+(A[2]-C[2])^2); c := sqrt((B[1]-A[1])^2+(B[2]-A[2])^2);
                            a := 19
                                 (1/2)
                         b := 205     
                                 (1/2)
                         c := 6 2     
p := 1/2*(a+b+c); Aire := sqrt(p*(p-a)*(p-b)*(p-c)); r := evalf(Aire/p);
                     19   1    (1/2)      (1/2)
                p := -- + - 205      + 3 2     
                     2    2                    
                        r := 2.727070319
Ag := arccos((-a^2+b^2+c^2)/(2*b*c)); Bg := arccos((a^2-b^2+c^2)/(2*a*c)); Cg := arccos((a^2+b^2-c^2)/(2*a*b)); verif := evalf(Ag+Bg+Cg);
                              / 7     (1/2)  (1/2)\
             Ag := Pi - arccos|--- 205      2     |
                              \410                /
                                 1   
                           Bg := - Pi
                                 4   
                               /13     (1/2)\
                   Cg := arccos|--- 205     |
                               \205         /
                      verif := 3.141592655

#Calcul des rayons des 3 cercles
r1 := evalf((1/2)*r*(1+tan((1/4)*Bg))*(1+tan((1/4)*Cg))/(1+tan((1/4)*Ag)));
                       r1 := 1.190796377
r2 := evalf((1/2)*r*(1+tan((1/4)*Ag))*(1+tan((1/4)*Cg))/(1+tan((1/4)*Bg)));
                       r2 := 1.918607660
r3 := evalf((1/2)*r*(1+tan((1/4)*Bg))*(1+tan((1/4)*Ag))/(1+tan((1/4)*Cg)));
                       r3 := 2.244243923
xI := evalf((a*A[1]+b*B[1]+c . C[1])/(2*p)); yI := evalf((a*A[2]+b*B[2]+c . C[2])/(2*p));
                      xI := -0.4162698432
                       yI := 1.727070322
 interface(rtablesize = 10);
  kernelopts(version);
  Physics:-Version();
  with(geometry):
  with(plots):
  S := segment:
  L := line:
  Per := PerpendicularLine:
  R := 5: xA := 0: yA := 0:
  point(A, xA, yA):
  xI := (1/3)*R:
  yI := 0:
  point(I1, xI, yI):
  circle(C, [A, R]):
  quadri := proc (t)
                  local xM, yM, xN, yN, xE, yE, dr1;
                  xM := evalf(R*cos(t));
                  yM := evalf(R*sin(t));
                  point(M, xM, yM);
                  L(lMI, [M, I1]);
                  intersection('h', C, lMI, [M, N]);
                  L(lAM, [A, M]);
                  L(lAN, [A, N]);
                  Per(lME, M, lAM);
                  Per(lNE, N, lAN);
                  intersection(E, lME, lNE);
                  S(sAM, [A, M]);
                  S(sAN, [A, N]);
                  S(sME, [M, E]);
                  S(sNE, [N, E]);
                  dr1 := draw({sME, sNE, lMI(color = blue), sAM(color = black), sAN(color = black)});
                  display({dr1})
            end proc:
  dr := draw({C}, view = [-6 .. 17, -10 .. 6]);
  display([dr, quadri(.7), quadri(1), quadri(1.2)], view = [-6 .. 17, -10 .. 6]);
  xA := -1;
  yA := 5;
  xB := -1; # Observation: yB is undefined but is used
            # later. Is this an issue?? I have included
            # a "random" value for yB in the next line here
  yB:= 2;
  xC := -7;
  yC := -1;
  _EnvHorizontalName := x;
  _EnvVerticalName := y;
  point(A, xA, yA);
  point(B, xB, yB);
  # point(C, C[1], C[2]); # C[1]and C[2] are undefined
                          # maybe you meant xC and yC
                          # I have changed this in the
                          # following
  point(C, xC, yC);
  coordinates(B);
  triangle(Tr, [A, B, C]);
  whattype(A); # type() generally takes two arguments
           # Maybe you intended to use whattype()??
#
# None of A[1], A[2]], [B[1], B[2]], [C[1], C[2]]
# are defined by the above code. No idea how you expect the
# following to work - Maybe you intended xA, yA, xB, yB, xC
# and yC????
#  
#  Po := pointplot([[A[1], A[2]], [B[1], B[2]], [C[1], C[2]], [xI, yI]],
#                   color = blue,
#                   axes = none,
#                   view = [-7 .. 12, -1 .. 5]
#                 );
  Po := pointplot([[xA, yA], [xB, yB], [xC, yC], [xI, yI]],
                   color = blue,
                   axes = normal,
                   symbol=solidcircle,
                   symbolsize=20,
                   view = [-7 .. 12, -1 .. 5]
                 );
                               10
  Maple 2018.2, X86 64 WINDOWS, Nov 16 2018, Build ID 1362973
"D:\lib\update.mla", 2018, November 16, 9:40 hours, version in 

   the MapleCloud: unable to determine, version installed in 

   this computer: not installed
Error, (in geometry:-point) the first argument is expected of type name
Error, (in geometry:-circle) wrong type of arguments
Error, (in geometry:-draw) unknown geometric object  [12, -1]
Error, (in geometry:-intersection) wrong type of arguments
                            xA := -1
                            yA := 5
                            xB := -1
                            yB := 2
                            xC := -7
                            yC := -1
                    _EnvHorizontalName := x
                     _EnvVerticalName := y
Error, (in geometry:-point) the first argument is expected of type name
Error, (in geometry:-point) the first argument is expected of type name
Error, (in geometry:-point) the first argument is expected of type name
Error, (in geometry:-coordinates) wrong type of argument
Error, (in geometry:-triangle) wrong type of arguments
                              list

 

with(geometry);
point(A, A[1], A[2]);
Error, (in geometry:-point) the first argument is expected of type name
Examples
with(geometry);
point(B, 2, 0);
Error, (in geometry:-point) the first argument is expected of type name
form(B);
                              FAIL
coordinates(B);
Error, (in geometry:-coordinates) wrong type of argument
HorizontalCoord(B);
Error, (in geometry:-HorizontalCoord) wrong type of argument
VerticalCoord(B);
Error, (in geometry:-VerticalCoord) wrong type of argument
detail(B);
Error, (in geometry:-detail) unknown object:  -7

 

The code you gave me work in certain cas but not in the other. I can't understand why.

interface(rtablesize = 10); kernelopts(version); Physics:-Version(); with(geometry); with(plots); S := segment; L := line; Per := PerpendicularLine; R := 5; xA := 0; yA := 0; point(A, xA, yA); xI := (1/3)*R; yI := 0; point(I1, xI, yI); circle(C, [A, R]); quadri := proc (t) local xM, yM, xN, yN, xE, yE, dr1; xM := evalf(R*cos(t)); yM := evalf(R*sin(t)); point(M, xM, yM); L(lMI, [M, I1]); intersection('h', C, lMI, [M, N]); L(lAM, [A, M]); L(lAN, [A, N]); Per(lME, M, lAM); Per(lNE, N, lAN); intersection(E, lME, lNE); S(sAM, [A, M]); S(sAN, [A, N]); S(sME, [M, E]); S(sNE, [N, E]); dr1 := draw({sME, sNE, lMI(color = blue), sAM(color = black), sAN(color = black)}); display({dr1}) end proc; dr := draw({C}, view = [-6 .. 17, -10 .. 6]); display([dr, quadri(.7), quadri(1), quadri(1.2)], view = [-6 .. 17, -10 .. 6]);
  Maple 2018.2, X86 64 WINDOWS, Nov 16 2018, Build ID 1362973
"D:\lib\update.mla", 2018, November 16, 9:40 hours, version in 

   the MapleCloud: unable to determine, version installed in 

   this computer: not installed
Error, (in geometry:-point) the first argument is expected of type name
Error, (in geometry:-circle) wrong type of arguments
Error, (in geometry:-draw) unknown geometric object  [12, -1]
Error, (in geometry:-intersection) wrong type of arguments

with(geometry); xA := -1; yA := 5; xB := -1; xC := -7; yC := -1;
_EnvHorizontalName := x;

_EnvVerticalName := y;

point(A, xA, yA); point(B, xB, yB); point(C, C[1], C[2]);
Error, (in geometry:-point) the first argument is expected of type name
Error, (in geometry:-point) the first argument is expected of type name
Error, (in geometry:-point) the first argument is expected of type name
coordinates(B);
Error, (in geometry:-coordinates) wrong type of argument
triangle(Tr, [A, B, C]);
Error, (in geometry:-triangle) wrong type of arguments
line(LAB, [A, B]); line(LBC, [B, C]); line(LAC, [A, C]);
Error, (in geometry:-line) wrong type of arguments
Error, (in geometry:-line) wrong type of arguments
Error, (in geometry:-line) wrong type of arguments
type(A);
                             false

Po := pointplot([[A[1], A[2]], [B[1], B[2]], [C[1], C[2]], [xI, yI]], color = blue, axes = none, view = [-7 .. 12, -1 .. 5]);
NULL;

display(Po);

Thank you very much for your code.
I am a 80 years old fan  of math. and I am trying to solve old problems of 1945 French syllabus using Maple with difficulties.

I am sorry I make a mistake, I meant triangle (MPQ) in stead of (MNP).
How animate correctly this geometrical figure ?

with(plots):with(geometry): unprotect(O):unprotect(D):unprotect(I) R:= 5: alpha:= (1/9)*Pi: beta:= (1/3)*Pi: n:= 100: dt:= 2*Pi/n: C1:= plot([R*cos(t), R*sin(t), t = 0 .. 2*Pi], color = blue): O:= [0, 0]: M:= [R*cos(beta), R*sin(beta)]: A:= [R*cos(alpha), R*sin(alpha)]: B:= [R*cos(alpha+Pi), R*sin(alpha+Pi)]: AB:= plot([A, B]): OM:= plot([O, M]): P:= [R*cos(t0*dt), R*sin(t0*dt)]: Q:= [R*cos(dt*t0+Pi), R*sin(dt*t0+Pi)]: point(M,M[1],M[2]):point(P,P[1],P[2]):point(Q,Q[1],Q[2]): triangle(T,[M,N,P]):bisector(bM, M, T):ExternalBisector(exbM,M,T): line(lMP, [M,P]):line(lMQ, [M,Q]):line(lPQ,[P,Q]):line(lAB,[A,B]): intersection(D,bM,lPQ):intersection(E,bM,lAB): intersection(Dp,exbM,lPQ):intersection(Ep,exbM,lAB): circle(Cir,[D,M,Dp]):center(I,Cir): diamob:= seq ( display ( [ plot( [P, Q], color = green), plot([M, P]), plot([M, Q]) ] ), t0 = 0 .. n ): tp:= textplot( [ [A[1]+.3, A[2], "A"], [B[1]-.3, B[2], "B"], [M[1]+.3, M[2]+.3, "M"] ] ): display( [ AB, C1, tp, display ( diamob,Cir,I, insequence = true ) ], scaling = constrained ); Error, missing operator or `;` interface(rtablesize=10): Too many errors. Thank you

Procedure Tunnell don't give a good answer for 24 and 41 (numbers <100)
Thank you

restart; with(NumberTheory);

Tunnell := proc (n) local a, b, A, B, C, Dd, k; if not IsSquareFree(n) then return false end if; if `mod`(n, 8) = 5 or `mod`(n, 8) = 6 or `mod`(n, 8) = 7 then if `mod`(n, 2) = 1 then A := 0; B := 0; for a from ceil(-sqrt((1/2)*n)-1) to ceil(sqrt((1/2)*n)+1) do for b from ceil(-sqrt(n)-1) to ceil(sqrt(n)+1) do k := n-2*a^2-b^2; if is(sqrt((1/32)*k), integer) then A := A+1 else if is(sqrt((1/8)*k), integer) then B := B+1 end if end if end do end do; if 2*A = B then return true else return false end if elif `mod`(n, 2) = 0 then C := 0; Dd := 0; for a from ceil(-sqrt((1/8)*n)-1) to ceil(sqrt((1/8)*n)+1) do for b from ceil(-sqrt((1/2)*n)-1) to ceil(sqrt((1/2)*n)+1) do k := (1/2)*n-4*a^2-b^2; if is(sqrt((1/32)*k), integer) then C := C+1 else if is(sqrt((1/8)*k), integer) then Dd := Dd+1 end if end if end do end do; if 2*C = Dd then return true else return false end if end if else return false end if end proc;
Tunnell(5);
Tunnell(34);
Tunnell(55);
Tunnell(61);
Tunnell(41);

Tunnell(79);
select(Tunnell, [seq(k, k = 1 .. 100)]);
Unperfect procedure (found on internet) which is not too bad. How to improve it ?

Thank you for tomleslie's explanations.
Some remarks : OEIS fondation in A042963, A003273 give Mathematica code for decomposition of congruent numbers ;
                           magma or SageMath give answers very quickly for such problem using elliptic curves. And Maple ?

From a given number u who is congruent how to find the rational sides of rectangle triangle of which u is the area.
Example from 34 I would like 24,17/6 et 154/6 for 24*17/6*1/2=34 ;
               fom 111 I would like 444/35, 35/2 et 1513/70 for 444/35*35/2*1/2=111.
Thanks you.

There is a problem this procedure FC does not work

I don't see what is u ? Thank you.