@ Kitonum, your command is not correct with the equation x^4 -(3*m+2)*x^2 + 3*m+1 = 0.

Thank for your answer. I am sorry about my question. I see, I asked many quetions, but, there are problems I can not solve them by myself, therefore I posted at here.  Please forgive me and help me. 

Thank for your answer. I am sorry about my question. I see, I asked many quetions, but, there are problems I can not solve them by myself, therefore I posted at here.  Please forgive me and help me. 

Thank you very much. How to get all solution? 

Thank you very much. How to get all solution? 

Thank you very much

Thank you very much

I tried

restart;

L:=[]:

k:=10:

for x1 from -k to k  do

for y1 from -k to k  do

for x2 from -k to k  do

for y2 from -k to k  do

for x3 from -k to k  do

for y3 from -k to k do

G:=[1,1]:

H:=[3,3]:

if x1+x2+x3=3*G[1] and y1+y2+y3=3*G[2] and (H[1]-x1)*(x3-x2) +(H[2]-y1)*(y3-y2) = 0 and (H[1]-x2)*(x2-x1) +(H[2]-y2)*(y2-y1)=0 and (x2-x1)*(y3-y1) +(x3-x1)*(y2-y1) <>0 then

L:=[op(L), [[x1,y1], [x2,y2], [x3,y3]]] fi;

od: od: od: od: od: od:

nops(L);  L;

I tried

restart;

L:=[]:

k:=10:

for x1 from -k to k  do

for y1 from -k to k  do

for x2 from -k to k  do

for y2 from -k to k  do

for x3 from -k to k  do

for y3 from -k to k do

G:=[1,1]:

H:=[3,3]:

if x1+x2+x3=3*G[1] and y1+y2+y3=3*G[2] and (H[1]-x1)*(x3-x2) +(H[2]-y1)*(y3-y2) = 0 and (H[1]-x2)*(x2-x1) +(H[2]-y2)*(y2-y1)=0 and (x2-x1)*(y3-y1) +(x3-x1)*(y2-y1) <>0 then

L:=[op(L), [[x1,y1], [x2,y2], [x3,y3]]] fi;

od: od: od: od: od: od:

nops(L);  L;

@ Kitonum, the coordinates H(3,3). Please help me.

@ Kitonum, the coordinates H(3,3). Please help me.

> restart:

with(geom3d):

point(A,1/3,5/7,8/5):

point(B,-1/2,2/5, -1/8):

point(C,1,-1,-4):

plane(P,[A,B,C],[x,y,z]):

n:=NormalVector(P);

cdn:=ilcm(op(denom~(n)));

n:=cdn*n;

k:=igcd(n[1],n[2],n[3]):

if n[1]<>0 then

sign(n[1])*cdn/k*sort(Equation(P))

elif n[2] <> 0 then sign(n[2])*cdn/k*sort(Equation(P))

else sign(n[3])*cdn/k*sort(Equation(P))

end;

> restart:

with(geom3d):

point(A,1/3,5/7,8/5):

point(B,-1/2,2/5, -1/8):

point(C,1,-1,-4):

plane(P,[A,B,C],[x,y,z]):

n:=NormalVector(P);

cdn:=ilcm(op(denom~(n)));

n:=cdn*n;

k:=igcd(n[1],n[2],n[3]):

if n[1]<>0 then

sign(n[1])*cdn/k*sort(Equation(P))

elif n[2] <> 0 then sign(n[2])*cdn/k*sort(Equation(P))

else sign(n[3])*cdn/k*sort(Equation(P))

end;

I see, the line r:=convert(domain, list); should be repair r:=convert(Domain, list);

I see, the line r:=convert(domain, list); should be repair r:=convert(Domain, list);