**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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