#1: 

with(plots):
with(plottools):
Diag:=proc(a,b) local x1, A, B, AB, C,x,y, labelA, labelB, labelAB, labelC, lineAandB, lineAB0:

x1:=implicitplot(y^2 = x^3 + a*x + b);

A:=RP(a,b);
A:=pointplot(A):
labelA:=textplot([op(A), 'A'], 'align'={'above'});
B:=RP(a,b);
B:=pointplot(B):
labelB:=textplot([op(B), 'B'], 'align'={'above'});
AB:=AddE(a,b,A,B);
AB:=pointplot(AB):
labelAB:=textplot([op(AB), 'A+B'], 'align'={'above'});

#lineAandB is the line between A and B using point slope form
lineAandB:= y=(B[2]-A[2])/(B[1]-A[1])*(x-A[1])+A[2];

#lineAB0 is the vertical line through A+B
lineAB0:= x=AB[1];

C:=solve({lineAandB, lineAB0}, {x,y});
C:=pointplot(C):

#the intersection of these lines is precisely the point C
labelC:=textplot([op(C), 'C'], 'align'={'above'});

#we display all relevant points and labels along with the elliptic curve
display({x1,A,B,AB,C,labelA, labelB, labelAB, labelC}, 'view'=[-10..10, -20..20]);

end:

print(`made it here`);

#2: 

IntSol:=proc(a,b,K) local y,x, S,t:

S:=isolve({y^2-x^3+a*x-b});

select(t-> eval(x<K,-x<K, t),{S});

end: 
#Credit:  I adapted info from this thread to include the condition #that x must be between -K and K https://mapleprimes.com/questions/#100754-Using-Isolve-With-Conditions


#3:

SolChain:=proc(a,b,S1,S2,K) local L : 

#initializing the list
L:=[S1, S2];

while i<=K+2 do

L:=[ops(L), AddE(a,b,L[-2],L[-1])];
i:=i+1;

od:

end:

#I think we could keep adding a point infinitely many times and keep getting new points
#We aren't in a finite field, so we don't have to worry about coming back.
#I believe we could get countably infinitely many points this way. 


#_____________________

#C20.txt: April 1, 2024
Help:=proc(): print(`RP(a,b), DoesLie(a,b,P), AddE(a,b,P,Q) `):end:
#Elliptic curve: y^2=x^3+a*x+b 
RP:=proc(a,b) local x: x:=rand(0..200)()/100: [x,sqrt(x^3+a*x+b)]:end:

DoesLie:=proc(a,b,P): P[2]^2-P[1]^3-a*P[1]-b:end:

AddE:=proc(a,b,P,Q) local x,y,s,eq,sol,R,xR:
 s:=(P[2]-Q[2])/(P[1]-Q[1]):
  y:=P[2]+(x-P[1])*s:
sol:={solve(y^2-x^3-a*x-b,x)};


if not (nops(sol)=3 and member(P[1],sol) and member(Q[1],sol)) then
  print(`Something bad happened`):
  RETURN(FAIL):
fi:

sol:=sol minus {P[1],Q[1]}:
xR:=sol[1]:
[xR,-sqrt(xR^3+a*xR+b)]:

end: