Archimedes broken chord theorem
Source: ISI(BS) 2006 #4
June 13, 2012
geometryangle bisectorgeometry proposed
Problem Statement
In the figure below, is the midpoint of the arc and the segment is perpendicular to the chord at . If the length of the chord is , and that of the segment is , determine the length of in terms of .
[asy]
unitsize(1 cm);
pair A=2dir(240),B=2dir(190),C=2dir(30),E=2dir(135),D=foot(E,B,C);
draw(circle((0,0),2)); draw(A--B--C); draw(E--D); draw(rightanglemark(C,D,E,8));
label("",A,.5A); label("",B,.5B); label("",C,.5C); label("",E,.5E); label("",D,dir(-60));
[/asy]