Problem

1963 AHSME Problem 18

Chord $EF$ is the perpendicular bisector of chord $BC$ , intersecting it in $M$ . Between $B$ and $M$ point $U$ is taken, and $EU$ extended meets the circle in $A$ . Then, for any selection of $U$ , as described, $\triangle EUM$ is similar to:

[asy] pair B = (-0.866, -0.5); pair C = (0.866, -0.5); pair E = (0, -1); pair F = (0, 1); pair M = midpoint(B--C); pair A = (-0.99, -0.141); pair U = intersectionpoints(A--E, B--C)[0]; draw(B--C); draw(F--E--A); draw(unitcircle); label("$B$", B, SW); label("$C$", C, SE); label("$A$", A, W); label("$E$", E, S); label("$U$", U, NE); label("$M$", M, NE); label("$F$", F, N); //Credit to MSTang for the asymptote[/asy]

$\textbf{(A)}\ \triangle EFA \qquad \textbf{(B)}\ \triangle EFC \qquad \textbf{(C)}\ \triangle ABM \qquad \textbf{(D)}\ \triangle ABU \qquad \textbf{(E)}\ \triangle FMC$

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

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Category: AHSME
Points: 2
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