A circle of Apollonius with respect to given foci O and P and real number r is the locus of points B with BO/BP = r. The upper half of the image diagrams the proof.

Any circle through the foci O and P will be orthogonal to the circle of Apollonius. The lower half of the image diagrams the proof.

Note that the green and red angles are bisected by the segments through A.

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