HYACINTHOS 6469

Dear Floor van Lamoen, Paul Yiu and other Hyacinthians,

Let me put together some old remarks:

1. (Floor van Lamoen in message #4547)

Let A'B'C' be the reflections of ABC through a point D, then the
circumcircles of AB'C', A'BC' and A'B'C concur in a point.

2. (Paul Yiu in message #4548)

The point of concurrence is the fourth intersection of the
circumcircle with the circumconic with center D.

3. (Reduced to elementary version of 2.)

The point of concurrence lies on the circumcircle of triangle ABC.

4. (J. R. Musselman, see message #1026)

If D is the nine-point center of triangle ABC, then the point of
concurrence is the Feuerbach point of the tangential triangle of ABC.

ARE THERE ANY SYNTHETIC PROOFS? (for 1., 3., 4.)

Sincerely,
Darij Grinberg