It is known that the parallel axiom has been proven independent of Euclidean geometry in the sense that its validity or falsity alike does not affect the validity of Euclidean geometry, but I have succeeded in proving its validity in Euclidean geometry, and I am not sure of the validity of the proof or its falsity. Is there a fallacy in the proof such as Incognito circular reasoning, or is the evidence valid? I'll lay out the proof now:
Parallel lines do not intersect no matter how long they are Proof:
$L⊥V$ in $A$, and $T⊥V$ in $B$, and therefore $L∥T$ because the two perpendiculars are parallel. We know that for two non-separating lines if they intersect they will intersect at exactly one point, so if $L,T$ intersect at point $M$ then it will be $M∉V$. We choose $P$ a random point from $V$, we note that quadrilateral $APBM$ contains two opposite right angles, so $APBM$ is a circular quadrilateral, for three points located on the same line, the circle passing through them is the same as the straight through them (the straight is a special case of the circle where the length of the radius is infinite ), so the circle passing through points $A,P,B$ is the same as straight $V$, and since $APBM$ is a circular quadrilateral, the circle passing through points $A,P,B$ must also pass through the fourth point $M$, and therefore $M∈V$. This is a contradiction, since $M∈V$ and $M∉V$ cannot be at the same time, and thus the assumption that $L,T$ intersect at point $M$ is wrong.