Quote:

Given three lines $a, b, c$ and two vectors $\vec{u}, \vec{v}$ , 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 $\vec{v}$ .

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

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