From San Gaku to Special Relationship?

In the large circle centered at $O,$

• both $\overline{AB}$ and $\overline{CD}$ are its diameters passing through $O;$
• the small, orange circle is tangent to $\overline{AB}$ at $P,$ $\overline{CD}$ at $Q,$ and circle $O$ at $R;$
• $\overline{RQ}$ and $\overline{RP}$ are extended to meet circle $O$ at $X$ and $Y,$ respectively.

True or False?

For any radius of circle $O$ and the diameters configuration, $|AC| = |XY|.$