1954 AHSME Problems/Problem 39
The locus of the midpoint of a line segment that is drawn from a given external point 
to a given circle with center 
and radius 
, is:
Solution
Note that the midpoint of to the point 
is the image of under a homothety of factor 
with center . Since homotheties preserve circles, the image of the midpoint as varies over the circle is a circle centered at the midpoint of and the original center and radius half the original radius. Therefore, our answer is 
, and we are done.
\begin{asy}
import graph;
unitsize(60);
pair P, Q;
path c;
P = (2,0);
Q = dir(142);
c = Circle((0,0), 1, 100);
dot(P, red);
dot(Q, darkgreen);
draw(P--Q, red);
draw(c, darkgreen);
label("", P, N);
label("", Q, NW);
pair M;
M = (P+Q)/2;
dot(M, blue);
draw(Circle(P/2, 1/2, 100), blue);
label("
", M, SE);
\end{asy}
See Also
| 1954 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 35 |
Followed by Problem 37 | |
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| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
