Triangles

Lines of triangles

Given are three mutually intersecting lines a, b, c at points D_ab, D_bc, D_ca and two points on each (A,A₁), (B,B₁) and (C,C₁) respectively. For any real number x, construct the points A_x = (1-x)*A + x*A₁, B_x = (1-x)*B + x*B₁ and C_x = (1-x)*C + x*C₁. Triangle A_xB_xC_x depends on x, and has the following properties.
The sides of triangle A_xB_xC_x satisfy certain conditions studied in LinesOfSegments.html and LinesOfSegments2.html .
Triangle A_xB_xC_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_xB_xC_x move on lines tangent to the same parabola with the side.
[2] The medians of triangle A_xB_xC_x envelope respectively three other parabolas.
[3] The centroid of A_xB_xC_x moves on a line simultaneously tangent to the three parabolas of [2].

Lines of triangles

See Also