# Six Concyclic Points II

What Is This About?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny### Explanation

The applet attempts to introduce the following problem:

This is Problem 10710 from the *American Mathematical Monthly* (1999) proposed by Bogdan Suceava.

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A solution by Achilleas Sinefacopoulos was published next year (*Am Math Monthly*, 107, p. 572) and, as a theorem, it appeared in [Suceava and Yiu, p. 191].

Since BC||YZ, ∠EDC = ∠EZA. In addition,

In ΔEYZ, the median from E to YZ equals half of the latter, implying that the triangle is right: _{a} is the orthocenter of that triangle. It follows that DH_{a} is the third altitude and, since BC||YZ, _{a} ⊥ BC._{a}. Further, the right angle DEH_{a} (with E and D on the incircle) needs to be subtended by a diameter, placing H_{a} on that circle and making H_{a} antipodal to D: _{a}.

Finally, the circle through F (the foot of an altitude), E (the foot of another altitude), and A (the midpoint of the third side) is necessarily the nine-point circle in ΔDYZ. Thus the circle passes through the midpoints E' and F' of the sides DZ and DY and the Euler point I on the altitude from D.

### References

- B. Suceava, P. Yiu,
__The Feuerbach Point and Euler lines__,*Forum Geometricorum*, Volume 6 (2006) 191-197.

### Nine Point Circle

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Copyright © 1996-2018 Alexander Bogomolny