## Lines of triangles

Given are three mutually intersecting lines a, b, c at points Dab, Dbc, Dca and two points on each (A,A1), (B,B1) and (C,C1) respectively. For any real number x, construct the points Ax = (1-x)*A + x*A1, Bx = (1-x)*B + x*B1 and Cx = (1-x)*C + x*C1. Triangle AxBxCx depends on x, and has the following properties.
The sides of triangle AxBxCx satisfy certain conditions studied in LinesOfSegments.html and LinesOfSegments2.html .
Triangle AxBxCx itself can be degenerate at most for two values of x. For each such value we get then a line simultaneously tangent to the three parabolas envelopping the sides of the triangle for the various x's (see second reference above, where the construction of such a parabola is carried out). Using the above references show that:
[1] The middles of the sides of AxBxCx move on lines tangent to the same parabola with the side.
[2] The medians of triangle AxBxCx envelope respectively three other parabolas.
[3] The centroid of AxBxCx moves on a line simultaneously tangent to the three parabolas of [2].