LinesOfSegments2

Lines of segments

Given are two intersecting (at point D) lines a, b and two points on each (A,A₁) and (B,B₁) respectively. For any real number x, construct the points A_x = (1-x)*A + x*A₁, B_x = (1-x)*B + x*B₁. The segment A_xB_x depends on x, and has the following properties.
[1] AA_x/A_xA₁ = BB_x/B_xB₁ for all x.
[2] Lines L_x = A_xB_x pass through a fixed point E, if and only if the cross-ratios: (A,A₁,A_x,D) = (B,B₁,B_x,D).
[3] The above condition is equivalent to the equality of ratios DA/AA₁ = DB/DB₁.
[4] In the usual case (in which [2] or its equivalent [3] is not true), circles (A_xB_xD) pass through a fixed point F.
[5] In the usual case lines A_xB_x are tangent to a parabola with focus at F. In particular, lines a and b are also tangent to that parabola. In this case also the following are true:
[6] The intersection points A', B' of the tangent at the vertex of the parabola with lines a, b are on the minimal circle (c) of the bundle of circles (A_xB_xD).
[7] The minimal length of A_xB_x occurs at A'B'.
[8] For every fixed k, the points C_x on A_xB_x, such that A_xC_x/C_xB_x = k, lie also on a tangent to the parabola and inversely any tangent to the parabola, defines on lines A_xB_x points C_x satisfying the above condition.

The proofs are consequences of the discussion done in ThalesParabola.html . [7] only is not handled there, but it follows easily from [4].
The map from line (a) to line (b) defined by A_x --> B_x preserves cross-ratios of four points. The parabola enveloping lines A_xB_x is a case of the Chasles-Steiner generation of conics. See Line_Homography.html for the general example.

Lines of segments

See Also