[alogo] A property of parabolas

For parabolas, the segments, cut out by three fixed tangents, intersected by a fourth, have a ratio independent of the position of the fourth tangent.

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The reason for this property, which is valid only for parabolas, among the conics, lies in the preservetion of the cross ratio, defined by four tangents of a conic, which are intersected by a fifth tangent. Here the three tangents are the lines [AG], [BF], [CE] and the fourth is the line at infinity (which is also tangent to the parabola). In this case the cross ratio of the four tangents, cut by a fifth (here [DA]), reduces to the ratio AB/BC, which is independent of the location of the fourth tangent at D. In particular if this is 1 for the tangent at a particular point D, then it is 1 for the tangent at every point of the parabola. For a picture of the general theorem on the cross ratio of four tangents to a conic look at the file FourTangentsCrossRatio.html .
This property of parabolas leads to an interesting picture, related to the medial lines of the sides of a triangle. Look at MedialParabola.html .

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