## Re: [EMHL] First 2010 locus

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• Dear Paul I think that a generalization can be: Let ABCD be a quadrilateral and P a point. Which is the locus of P such that the circumcenters of
Message 1 of 3 , Jan 1, 2010
Dear Paul

I think that a generalization can be:

Let ABCD be a quadrilateral and P a point.
Which is the locus of P such that the circumcenters of
PAB,PBC,PCD,PDA are concyclic?

By using triangle geometry:

We take ABC as reference triangle. Let the h. coordinates of
the fixed point D be (p:q:r), and of the variable point P be
(x:y:z). Then we find the equation of the locus in terms of
the fixed p,q,r (and fixed elements of ABC).

Note that we can take both D,P as variable points. For example:
Let ABC be a triangle and P, P* two isogonal conjugate points
(we took D = P*) etc

> [APH]: Let ABC be a triangle, P a point, and O,Oa,Ob,Oc
> the circumcenters of
> ABC,PBC,PCA,PAB, resp. Which is the locus of P such
> that O,Oa,Ob,Oc are concyclic?
>
> [PY] This is the case if and only if P is the on the circumcircle
> of ABC. The four circumcenters coincide.

Happy New Year

APH
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