@Carl Love  I downloaded Maple 2016 trial (15 days). When I ran your code, I got the 
message Error, invalid `$` operator. I don't understand why. 

PS. How can I use your code with squares on circle? 

@Carl Love Thank for your help. I did.

@Carl Love I used Maple 18, but I got the 

Error, (in unknown) invalid input: add expects 2 arguments, but received 1

@Carl Love Thank you very much.

@Carl Love My version is Maple 2015. Therefore, I cann't compile your code. 

@Kitonum How can I see the order of vertices of square? Because, perhalps you sorted the line

Squares:=select(p->nops(op~(p))=12, L); 

@Kitonum When I ran all the results, I got "Error, invalid subscript selector". I think, because there are some lines are parallel. How can I compile all the results?

I want to write this tangent of equations in the simplest form. Example if the equation has the form -2x + 3y + 2 = 0. We write 2x - 3y - 2 = 0. Or If we have -3x + 6y - 9 = 0, we write x - 2y + 3 = 0. I tried with this code.

restart;
with(geometry): 
point(A, 1, -1): 
point(B, 3, 5):
Equation(line(d, [A, B], [x, y]));
d1 := lhs(Equation(line(d, [A, B], [x, y]))):
a := coeff(d1, x):
b := coeff(d1, y):
k := igcd(a, b):
if a <> 0 then sign(a)*sort(Equation(d))/k else sign(b)*sort(Equation(d))/k end;

How can I add this code into Perpendicularity Criterion?

@Carl Love Is your code right?

I think, true code is isolve({x > 3/2, x < 5/2})

@Markiyan Hirnyk Mathematica solve the equation with integer soluttions very good.

Reduce[{-13/74 < 7/37*f - 2/37*g, -1/2 < 2/37*f + 7/37*g, 

2/37*f + 7/37*g < -33/74,
7/37*f - 2 > 7*g < 1/2, -100 <= f <= 100, -100 <= g <= 100}, {f,
g}, Integers]

We get

(f == 2 && g == -3) || (f == 5 && g == -4) || (f == 9 &&
g == -5) || (f == 12 && g == -6) || (f == 16 &&
g == -7) || (f == 19 && g == -8) || (f == 23 &&
g == -9) || (f == 26 && g == -10) || (f == 30 &&
g == -11) || (f == 33 && g == -12) || (f == 37 &&
g == -13) || (f == 40 && g == -14) || (f == 44 &&
g == -15) || (f == 47 && g == -16) || (f == 51 &&
g == -17) || (f == 54 && g == -18) || (f == 58 &&
g == -19) || (f == 61 && g == -20) || (f == 65 &&
g == -21) || (f == 68 && g == -22) || (f == 72 &&
g == -23) || (f == 75 && g == -24) || (f == 79 &&
g == -25) || (f == 82 && g == -26) || (f == 86 &&
g == -27) || (f == 89 && g == -28) || (f == 93 &&
g == -29) || (f == 96 && g == -30) || (f == 100 && g == -31)

@vv I am sorry about my question not clear. One must find the parameters for which the equation has always two integer solutions.

@vv In my code of Mathematica, the equation has two solutions. 

@Kitonum Thank you very much. I have many problems about integer points. 

@Kitonum Thank you very much.

@Kitonum If I change 

to k:=combine((sin(x+pi/3))^2); How can I get the simplest solution?