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I understand that the blue and red lines are conjugate diameters; but can $GG'$ and $HH'$ be conjugate diameters, in the blue ellipse below? (G' and H' are symmetry points of G, H, by M.)

ABCD is a parallelogram and the blue ellipse is inscribed in ABCD, where each tangent points divide each segment by m : n internally.

enter image description here

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Yes, they are conjugate diameters. There's a theorem stating that all midpoints of chords parallel to a diameter of an ellipse lie on the conjugate diameter. $PQ$ is parallel to $MH$ by the intercept theorem, and $N$ is the midpoint of $PQ$ by triangle similarity. Hence the diameter conjugate to $HH'$ lies on line $MN$.

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