Find the radius of circle $P$ 
$ABCD$ is a rectangle with $AB = CD = 2$. A circle centered at $O$ is tangent to $BC, CD,$ and $AD$ (and hence has radius $1$). Another circle, centered at $P$, is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$. If line $AT$ is tangent to both circles at $T$, find the radius of circle $P$.

Can someone create a picture for this? I am having a hard time seeing how the circle centered at $P$ can be tangent to the circle centered at $O$.

This is too long for a comment. 
Let us assume that @Briantung ‘s diagram is correct. If we identity the lengths of the line segments starting from D (in anti-clockwise direction), we will have the following diagram.
 
Then,  $1 + 2x + y = BC = AD = 1 + 2 – y$.
This implies $x + y = 1$ ......... (*) 
Now let draw some more lines (like MV and SR) to analyze the lengths of some segments. (Please refer to the second diagram.)

If the (*) is true, NS has to be y. Then, SU = x – y, RA = SB = SU + UB = (x – y) + y = x.
Thus, the tangent NM = NT = NU = x = RA.
Adopting this result, we construct the second circle as follows:-
1) Draw the line SR such that CDRS is a square of side 2.
2) Translate RA to MN such that RA = MN = x.
Then, NA should be our common tangent to the 2 circles with the point of contact at T. My third diagram clearly shows that it is not the case.

