### Construction of line knowing newton line direction

Source: AoPS

Given three lines *a, b, c* and two vectors **u, v**, construct a line *d*, with direction **u**, such that Newton line of the quadrilateral formed by the four lines *a, b, c, d * has direction **v**.

The three lines *a, b, c* form a triangle *ABC*. Let 𝒫 be the parabola inscribed in its medial triangle and whose axis has the direction of the vector **u**. Let *d* be the conjugate diameter of the direction of the vector **v**. 𝒫 and *d* intersects at a point *T*_{uv}. The tangent at *T*_{uv} to 𝒫 is Newton line of the quadrilateral *abcd*, which has the direction of the vector **v**.

Angel Montesdeoca. Enero, 2023