POLYA026:  Common chord of circles (proposed by Antreas P. Hatzipolakis, 2/4/02).

If AA', BB' are two intersecting at P chords of a circle, then the common chord of the circles with diameters AA', BB'
passes through P.

Discussion. [FvL, 2/7/02]:  The power of P w.r.t. the first circle is PA*PA'=PB*PB'. From that we see that the powers of P w.r.t. the circles with diameters AA' and BB' are equal. So P lies on the radical axis of these two circles, which is the line containing their common chord. Since P lies in the interior of both circles, P lies on the chord itself.
 

Bibliography.

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