From San Gaku to Special Relationship?

Geometry Level 3

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, \(|AC| = |XY|.\)

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