# 1965 AHSME Problems/Problem 4

## Problem

Line $\ell_2$ intersects line $\ell_1$ and line $\ell_3$ is parallel to $\ell_1$. The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is:

$\textbf{(A)}\ 0 \qquad \textbf{(B) }\ 1 \qquad \textbf{(C) }\ 2 \qquad \textbf{(D) }\ 4 \qquad \textbf{(E) }\ 8$

## Solution

$[asy] draw((-36,0)--(36,0), arrow=Arrows); label("\ell_2", (40,0)); draw((14,-32)--(30,32), arrow=Arrows); label("\ell_1", (32,36)); draw((-4,-32)--(12,32), arrow=Arrows); label("\ell_3", (14,36)); draw((5,-32)--(21,32), dotted, arrow=Arrows); label("\ell_4", (23,36)); [/asy]$

The lines are coplanar, $\ell_1 \parallel \ell_3$, and $\ell_1$ intersects $\ell_2$. Therefore, $\ell_3$ also intersects $\ell_2$. The locus of all points equidistant from parallel lines $\ell_1$ and $\ell_3$ is a third parallel line in between them. Let this line be $\ell_4$, and let the distance from $\ell_4$ to either $\ell_1$ or $\ell_3$ be $d$. The points equidistant from lines $\ell_1$, $\ell_2$, and $\ell_3$ must all lie on $\ell_4$ and be a distance $d$ from line $\ell_2$. There are only 2 points, on either side of $\ell_2$, which satisfy these conditions. Thus, our answer is $\fbox{(C) 2}$.