Given are three mutually intersecting lines a, b, c at points D_{ab}, D_{bc}, D_{ca} and two points on each (A,A_{1}), (B,B_{1}) and (C,C_{1}) respectively. For any real number x, construct the points A_{x} = (1-x)*A + x*A_{1}, B_{x} = (1-x)*B + x*B_{1} and C_{x} = (1-x)*C + x*C_{1}. Triangle A_{x}B_{x}C_{x} depends on x, and has the following properties. The sides of triangle A_{x}B_{x}C_{x} satisfy certain conditions studied in LinesOfSegments.html and LinesOfSegments2.html . Triangle A_{x}B_{x}C_{x} 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 A_{x}B_{x}C_{x} move on lines tangent to the same parabola with the side. [2] The medians of triangle A_{x}B_{x}C_{x} envelope respectively three other parabolas. [3] The centroid of A_{x}B_{x}C_{x} moves on a line simultaneously tangent to the three parabolas of [2].