Let there be two circles. Find all points $M$ such that there exist two points, one on each circle such that $M$ is their midpoint.
Problem
Source: Croatian MEMO Selection Test 2007
Tags: symmetry, ratio, geometry proposed, geometry
skuge
13.08.2007 03:50
First answer this question:
For two circles $ w_{1}$, $ w_{2}$, find all points $ M$ such that the central symmetry of $ w_{1}$ is tangent to $ w_{2}$
Altheman
15.08.2007 03:50
Let $ P$ be a point fixed on one circle, then the locus of $ M$ is the other circle dialated by ratio 1/2. Repeat this for all points and you get like a ring.
Amir.S
20.08.2007 18:10
let centers of circles be $ O_{1}$ and $ O_{2}$ and $ N$ be midpoint of $ O_{1}O_{2}$ and $ A$ be a point on $ (O_{1})$ and $ B$ be on $ (O_{2})$. it's easy to prove that:$ \overrightarrow{NM}=\overrightarrow{O_{1}A}+\overrightarrow{O_{2}B}$ the maximum lenght of $ MN$ is $ r_{1}+r_{2}$ when $ O_{1}A\parallel O_{2}B$ and minimum of $ MN$ is $ |r_{1}-r_{2}|$ when $ \overrightarrow{O_{1}A}$ and $ \overrightarrow{O_{1}B}$ be in opposite direction.hence the locus of $ M$ is the pink part in the figure.
Attachments:
