Ken's POTW


Crossing Chords
        +++++++++
     +++         +++
   +S               ++
  + |\                +
 +  | \                +
 +  |  \               +
+   |   \              _Q
+   |    \  O       _- /+
+   |     \      _-   / +
 +  |      \  _-     / +
 A--X-------C-------Y--B
  + |    _-  \     /  +
   ++ _-      \   / ++
    P+++       \ /++
        ++++++++R
In circle O, a chord AB is drawn, with midpoint C. Through C, two more chords are drawn, PQ and RS. Connecting the endpoints, PS passes through AC at point X, and QR passes through BC at point Y. Show that CX = CY.

First try showing:

  • If RS is a diameter of O, that (CR)(CS) = (CA)(CB)
  • that (CP)(CQ) = (CR)(CS).

Source: The rec.puzzles newsgroup long ago, though I can't find it in the rec.puzzles archive.


Solution
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