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Regarding your first issue, you could try:

sage: solve([circ, line], [x,y])
[[x == -1/2*sqrt(2), y == -1/2*sqrt(2)], [x == 1/2*sqrt(2), y == 1/2*sqrt(2)]]

Regarding your second issue, Sage reports the inequations because it is not able to solve the simple system of inequalities. Indeed, qepcad is able to decide that there is no solution:

sage: qf = qepcad_formula
sage: F = qf.and_([circ, badIneq]) ; F
[x^2 + y^2 = 1 /\ y > 1]
sage: qepcad(F, solution='extended')
FALSE
sage: qepcad(F)
FALSE

You should be able to install qepcad with the command (in a terminal)

sage -i qepcad

If this does not work, you should provide some informations on your system.

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No.2 Revision

Regarding your first issue, you could try:

sage: solve([circ, line], [x,y])
[[x == -1/2*sqrt(2), y == -1/2*sqrt(2)], [x == 1/2*sqrt(2), y == 1/2*sqrt(2)]]

Regarding your second issue, Sage reports the inequations because it is not able to solve the simple system of inequalities. Indeed, qepcad is able to decide that there is no solution:

sage: qf = qepcad_formula
sage: F = qf.and_([circ, badIneq]) ; F
[x^2 + y^2 = 1 /\ y > 1]
sage: qepcad(F, solution='extended')
FALSE
sage: qepcad(F)
FALSE

You should be able to install qepcad with the command (in a terminal)

sage -i qepcad

If this does not work, you should provide some informations on your system.

You could also have a look at sympy, which is installed in Sage: https://docs.sympy.org/latest/modules/solvers/inequalities.html