HYACINTHOS 21433

BW

> A possible generalization led to something curious. Take all circles that are
> tangent to two of the sidelines BC, CA, AB.  Four of these circles pass through
> the Feuerbach point, the incircle and 3 others. The centers of those 3 others
> are collinear.

  

These 3 circles meet at the Feuerbach point and at ( (b-c)^4 (b+c-a)^3, ... , ...)
 > Does this work if the Feuerbach point is replaced with some other point on the

> incircle?

I've solved this. If the Feuerbach point is changed to one of the 3 points where the incircle meets the sidelines, these are degenerate cases where there aren't 4 such circles. And the Feuerbach point is the only other point on the incircle where this collinearity holds.
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Barry Wolk