Similar Triangles
Source: AHSME 1963 Problem 18
January 9, 2014
Asymptotegeometryperpendicular bisectorAMC
Problem Statement
Chord is the perpendicular bisector of chord , intersecting it in . Between and point is taken, and extended meets the circle in . Then, for any selection of , as described, triangle is similar to triangle:[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, SW);
label("", C, SE);
label("", A, W);
label("", E, S);
label("", U, NE);
label("", M, NE);
label("", F, N);
//Credit to MSTang for the asymptote
[/asy]