# f(leftVarlist)== f(rigthVarList)=solve(sysEqu,leftVarList,rigthVarList)

I will find it convenient that SageMath would have a solve function like this:

sysEqu=[] # system of equations ; leftVarList=[] # left variables list ; rightVarList=[] # right variables list

solve(sysEqu,leftVarList,rigthVarList) It will return f(leftVarlist)== f(rigthVarList) if possible and return [] if not.

Maybe it is too hard to write this kind of function ? (I know simplification is difficult subject in math soft), or maybe the solve function is already able to do this, and I just need to set it up correctly ? (Sorry if it is the case!)

To illustrate my proposal I started to write a draft of this kind of function, (badly written, and which does not work correctly!) :

varL=[]
# all red segments
varL.append(var("A",latex_name=r"\color{red}{A}"))
varL.append(var("Ap",latex_name=r"\color{red}{A'}"))
varL.append(var("App",latex_name=r"\color{red}{A''}"))
# all green segments
varL.append(var("AppX",latex_name=r"\color{green}{\overline{A''X}}"))
varL.append(var("AU",latex_name=r"\color{green}{\overline{AU}}"))
varL.append(var("ApY",latex_name=r"\color{green}{\overline{A'Y}}"))
# all pink segments
varL.append(var("AX",latex_name=r"\color{pink}{\overline{AX}}"))
varL.append(var("ApW",latex_name=r"\color{pink}{\overline{A'W}}"))

# all blue segments
varL.append(var("X",latex_name=r"\color{blue}{X}"))
varL.append(var("U",latex_name=r"\color{blue}{U}"))
varL.append(var("W",latex_name=r"\color{blue}{W}"))
varL.append(var("Y",latex_name=r"\color{blue}{Y}"))

varL.append(var("Z",latex_name=r"\color{orange}{Z}"))

for v in varL :
assume(v,'real')
assume(v!=0)

eqL=[
AppX/X == AU/U,
AppX/X == ApY/Y,
App/X == A/U,
App/X == Ap/Y,
AppX/App == AU/A,
AppX/App == ApY/Ap,
A/X == Ap/W,
A/X == -(A - Ap)/(W - X),
A/AX == Ap/ApW,
X/AX == W/ApW,
AU/AX == ApY/ApW,
AU/(U - X) == -ApY/(W - Y),
AX/(U - X) == -ApW/(W - Y),
]

show(varL)

index=0
latexL=[]
for eq in eqL :
show(" equ # : ",index," \t : ",eq," \t valid =",eq.is_exact())
latexL.append(r"\text{equ #} "+str(index)+" : "+latex(eq))
index+=1

def isTheSame(t0,t1):
#test if tuple or list have same elements notwithstanding order
t0l=len(t0) ;t1l=len(t1)
if t0l!=t1l :
return False
for tt in t0:
if tt not in t1 :
return False
return True

# I have to tinker !!
# Do It YourSelf function which does not work correctly.

def solvit(eq0L,vare,mainVariables) :
advanticeVariablesL=[]
for eq in eq0L :
vart=eq.variables()
for v in vart :
if v not in mainVariables :
if v not in advanticeVariablesL :
advanticeVariablesL.append(v)
mainVariables=tuple(mainVariables)
advanticeVariables= tuple(advanticeVariablesL)
show("advantice Variables : ",advanticeVariables)
#                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              show("main :",mainVariables)

#stopLoop=False
#solution=0==0
for indexAdv in range(0,len(advanticeVariables)-1):
#if stopLoop:
#    break
v=advanticeVariables[indexAdv]
#show("var to remove : ",v," : ",indexAdv)
equLength=len(eq0L)-1
indexEq=0
while indexEq < equLength :
#if stopLoop:
#    break
#show("indexEq : ",indexEq)
if v  in (eq0L[indexEq]).variables(): # look for unwanted variables
sol=solve(eq0L[indexEq],v)
eq0L.remove(eq0L[indexEq])
equLength-=1
#show("eq0L length : ",equLength)
for indexEq in range(0,equLength-1):# replace v in all other equ
eqt=eq0L[indexEq].subs(sol).simplify_full()
eq0L[indexEq]=eqt
soleqt=solve(eqt,vare)
if len(soleqt)>0 :
#show("soleqt : ",soleqt)
if isTheSame(soleqt.variables(),mainVariables) :
solution=soleqt
#show("solution : ",solution)
return [solution,eq0L,indexAdv,advanticeVariables[indexAdv]]
#stopLoop=True
#break
indexEq+=1
return [0==1,eq0L,indexAdv,advanticeVariables[indexAdv]]
solution,remainingEquationsL,indexAdv,Adv=solvit(copy(eqL),W,[U,X,Y,W])

show("solution : ",solution," \t len remainingEquationsL : ",len(remainingEquationsL),
" \t step :",len(eqL)-len(remainingEquationsL),"\t index Adv :",indexAdv,"\t Adv : ",Adv)

# attempt with solve only

show("With solve only W  ",solve(eqL,W))
show("With solve [U,W,X,Y] ",solve(eqL,[U,W,X,Y]))
show("With solve only only W sol dict",solve(eqL,W,solution_dict=True))
show("With solve [U,W,X,Y] and sol dict",solve(eqL,[U,W,X,Y],solution_dict=True))


complete code on SageCell

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