# 1954 AHSME Problems/Problem 39

The locus of the midpoint of a line segment that is drawn from a given external point $P$ to a given circle with center $O$ and radius $r$, is: $\textbf{(A)}\ \text{a straight line perpendicular to }\overline{PO}\\ \textbf{(B)}\ \text{a straight line parallel to }\overline{PO}\\ \textbf{(C)}\ \text{a circle with center }P\text{ and radius }r\\ \textbf{(D)}\ \text{a circle with center at the midpoint of }\overline{PO}\text{ and radius }2r\\ \textbf{(E)}\ \text{a circle with center at the midpoint }\overline{PO}\text{ and radius }\frac{1}{2}r$

## Solution

Note that the midpoint of $P$ to the point $Q$ is the image of $Q$ under a homothety of factor $\frac{1}{2}$ with center $P$. Since homotheties preserve circles, the image of the midpoint as $Q$ varies over the circle is a circle centered at the midpoint of $P$ and the original center and radius half the original radius. Therefore, our answer is $\boxed{\text{(E)}}$, and we are done. $[asy] import graph; unitsize(60); pair P, Q; path c; P = (2,0); Q = dir(142); c = Circle((0,0), 1, 100); dot(P, red); dot(Q, darkgreen); draw(P--Q, red); draw(c, darkgreen); label("P", P, N); label("Q", Q, NW); pair M; M = (P+Q)/2; dot(M, blue); draw(Circle(P/2, 1/2, 100), blue); label("M", M, SE); [/asy]$

## See Also

 1954 AHSC (Problems • Answer Key • Resources) Preceded byProblem 35 Followed byProblem 37 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 All AHSME Problems and Solutions

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