Does the radius of the quadrant pass from the center of the inscribed circle?

Question

In the following picture:

The smaller circle is inscribed inside the quadrant, whose radius (OB) is 8. The original question (but not the question of this post) is that "find the radius of the inscribed circle and the area of shaded part". I managed to solve the problem using the guess that the line OT passes from the point P (P is the center of the inscribed circle). But I am not able to prove this. Can you give me a hint about how can I prove that OT passes the point P?

Answer 1

Notice that a tangent line passing through the point $T$ is tangent to both the quarter circle and the inscribed circle. That means the radii of the inscribed circle and the quarter circle are both perpendicular to the tangent line passing through the point $T$. That means the radius of the inscribed circle coincides with the radius from the quarter circle.

Answer 2

The great and the small circles have a common tangent in $T$ , so the orthogonal to this tangent passes through $P$ and $O$.