Nikolaos Dergiades, Harmonic conjugate circles relative to a triangle,
Forum Geometricorum, 12 (2012) 153--159.
Abstract.
We use the term harmonic conjugate conics, for the conics C, C* with equations
C: fx2+gy2+hz2+2pyz+2qz+2rxy = 0 and C*: fx2+gy2+hz2-2pyz-2qz-2rxy = 0, in
barycentric coordinates because if A_1, A_2 are the points where C meets the sideline BC of
the reference triangle ABC, then C* meets the same side at the points A_1', A_2' that are
harmonic conjugates of A_1, A_2 respectively relative to BC and similarly for the other sides of ABC.
So we investigate the interesting case where both C and C* are circles.
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