Let the tangent line passing through a point $A$ outside the circle with center $O$ touches the circle at $B$ and $C$. Let $[BD]$ be the diameter of the circle. Let the lines $CD$ and $AB$ meet at $E$. If the lines $AD$ and $OE$ meet at $F$, find $|AF|/|FD|$.
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tk1
21.01.2013 01:24
Let $FK$ be the perpendicular from $F$ to $EB.$ Because $\angle BDC=\angle BOC/2=\angle BOA,$ we conclude that the right triangles $\triangle ABO,\triangle EBD$ are similar and that $AO\parallel ED.$ Thus, $EA=AB.$ Therefore, in $\triangle EBD,$ $EO$ and $DA$ are medians and thus $EF/FO=2=EK/KB.\ (1)$ If we set $AK=y$ and $AB=x,$ we find from $(1)$ that $(x+y)/(x-y)=2$ and thus $y=x/3.$ Therefore, $AF/FD=AK/KB=y/(x-y)=1/2.$
sunken rock
21.01.2013 22:30
$BC$ is the altitude of the right-angled triangle $\Delta BDE$ and, easily, $A$ is the midpoint of $BE$, consequently $F$ is the centroid of $\Delta BDE$, a.s.o. Best regards, sunken rock
anamariaradu
26.03.2014 21:08
Or, we can see that ABOC is square and thus $AE/BE=1/2$. Using Menelaus theorem in the triangle $DAB$ with $O-F-E$ and we obtain $AF/FD=1/2$