An Archimedes Triangle

Consider the parabola $E \colon y = x^2$. A line through $T(1, 2)$ meets $E$ at two points $A$ and $B$. Let $l_1$ and $l_2$ be the tangent lines to $E$ at $A$ and $B$; $l_1$ meets the $x$-axis at $M$, $l_2$ meets the $x$-axis at $N$, and $l_1, l_2$ intersect at $P$.

1. Prove that $P$ lies on a fixed line. 2. If the area of triangle $PMN$ is $\sqrt{2}$, find the coordinates of $P$. 3. If $P, M, N, T$ lie on a common circle, find the coordinates of $P$.