## 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.

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 .