\documentclass[
	solutions,
	singlenumbering,
	nopagenumber
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

\title{Warm-Up: Travellers}
\uptitler{\smallurl{}}
\subtitle{Prepared by Mark on \today}

\begin{document}

	\maketitle

	\problem{}
	Four travellers are on a plane, each moving along a straight line at an arbitrary constant speed. \par
	No two of their paths are parallel, and no three intersect at the same point. \par
	We know that traveller A has met travelers B, C, and D, \par
	and that traveller B has met C and D (and A). Show that C and D must also have met. \par

	\begin{solution}
		When a body travels at a constant speed, its graph with respect to time is a straight line. \par
		So, we add time axis in the third dimension, perpendicular to our plane. \par
		Naturally, the projection of each of these onto the plane corresponds to a road.

		Now, note that two intersecting lines define a plane and use the conditions in the problem to show that no two lines are parallel.
	\end{solution}

\end{document}