Three collinear Circles and Simultaneous Diameters
Here we add a third proof to a problem addressed elsewhere:
Three circles centered at P, Q, R are concurrent at O and also meet pairwise at points D, E, and F.
Assume D, O, P are collinear and so are E, O, Q. Then F, O, R are also collinear.
This proof is due to Patrick Honner, a high school math teacher from New York.
The center lines PQ, QR, PR of the three circles are the perpendicular bisectors of the three common chords: FO, DO, and EO, respectively.
Assume, as stipulated in the problem, that P lies on the extension of OD and Q on the extension of OE. In ΔPQR,
This proof points to a possible generalization of the problem: instead of the common chords, we may consider the radical axes of two circles. Such a generalization is indeed valid and is treated on a separate page.
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