The relation between x, x' implies the relations: |xy|/|xz| = |x'y'|/|x'z'|, and |xy|/|x'y'| = |xz|/|x'z'|. Thus the subject reduces to the discussion initiated in Thales_General.html .

The determination of the focus F of the parabola and its subsequent construction is explained in ThalesParabola.html . The property of I to be the pole of the line-similarity-axis follows from the definition of the parabola. In the case the two lines are parallel (I goes to infinity) the parabola reduces to a point Q, all lines XX' going through that point. In that case the line-similarity-axis becomes the polar of Q with respect to the degenerate conic represented by the two parallel lines e and e'.

The real clue of the whole story is the property of the tangents of a parabola explained in ParabolaProperty.html .

The subject studied here gives another aspect of the theorem of Pappus discussed in PappusLines.html .

Chasles_Steiner_Envelope.html

HomographicRelation.html

LineHomographyAxis.html

PappusLines.html

ParabolaProperty.html

Thales_General.html

ThalesParabola.html

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