The following problem comes from Geometry in Conics by Akopyan and Zaslavasky.
Two travelers move along two straight roads with constant speeds. Prove that the line connecting them is always tangent to some parabola (the roads are not parallel and the travelers pass the intersection at different times).
How would one solve this problem? The construction seems hard to find.
It is not difficult to check (via $\Delta=0$) that the line through $(t,t)$ and $(t-1,1-t)$ is always tangent to the parabola having its focus at $(0,1)$ and its vertex at $\left(0,\frac{1}{2}\right)$. By applying a suitable affine map the claim follows.