Find an interesting Locus.

freemind

Find an interesting Locus.

**Posted:** Sun Mar 02, 2008 5:02 pm

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The fate of equilibrium is to end the eternity...

Erken

It is easy to see that $F$ is the nine-point circles center(easy and well-known fact).
So all we need to do is to find the locus of $A$ when $F$ -nine-point circle's center lies on $BC$ .
Then it is easy to see that $F\in BC\Longleftrightarrow AO\|BC$ .
Now the problem become obvious,if we take the point $M$ ,midpoint of $BC$ ,as $(0,0)$ ,then the locus of $A$ is hyperbola $\boxed{x^2-y^2=1}$ .
[Edited after reading freemind's private message.]

**Posted:** Sun Mar 02, 2008 6:49 pm

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A mathematician is a scientist who can figure out anything except such simple things as squaring the circle and trisecting an angle