Given three parallel lines on the plane. Find the locus of incenters of triangles with vertices lying on these lines (a single vertex on each line).
Problem
Source: I.F.Sharygin contest 2009 - Correspondence round - Problem 18
Tags: geometry, incenter, inradius, rectangle, geometry proposed
22.06.2009 16:29
Call $ r$ the parallel line that is contained in the stripe determined by the other two lines $ s,s'$. Clearly, if the incenter $ I$ is contained in the stripe defined by $ r$ and $ s$, then it must be closer to $ r$ than to $ s$; otherwise, since the incircle cannot touch line $ s$ because the triangle is entirely contained in the stripe between $ s$ and $ s'$, then the inradius is less than half the distance between $ r$ and $ s$, hence it would not cross $ r$, and any ray originating in any point on $ r$ and touching the incircle could not cross $ s'$. Similarly, if the incenter is contained in the stripe defined by $ r$ and $ s'$, then it is closer to $ r$ than to $ s'$. We will now show that for any point strictly inside the stripe defined by the parallel lines halfway between $ r$ and $ s$, and halfway between $ r$ and $ s'$, one point may be found on each of the three given lines such that this point is the incircle of the triangle thus formed. Take $ A\in r$, $ B\in s$ and $ C\in s'$, collinear and such that $ AB\perp r$. Displace $ C$ along $ s'$ away from $ A,B$. As $ C$ goes arbitrarily far, the figure tends towards an open rectangle with one side $ AB$, two sides being rays on $ r$ and $ s$, hence the incircle tends towards the circle touching $ AB$, $ r$ and $ s$, ie, the incenter tends towards a point on the line halfway between $ r$ and $ s$, getting arbitrarily close but never reaching it. From the original position, displace $ A$ along $ s$ away from $ B,C$, the incenter tends towards a point on the line halfway between $ r$ and $ s'$, getting arbitrarily close but never reaching it. Since $ C$ on the first displacement, and $ A$ on the second displacement, may be moved continuously, and in doing so the incenter moves continuously, for each line $ t$ parallel to $ r$ contained strictly in the stripe defined by the lines halfway from $ r$ and $ s$, and halfway from $ r$ and $ s'$, there is a position of $ A,B,C$ such that the incenter is on $ t$. Displace now simultaneously $ A,B,C$ along $ s,r,s'$ as needed until the incenter reaches any desired point on $ t$. The conclusion follows.