Six Concyclic Points II
What Is This About?
A Mathematical Droodle


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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

Explanation

The applet attempts to introduce the following problem:

Consider a triangle ABC with incenter I, the incircle touching the sides BC, CA, AB at D, E, F respectively. Let Y (respectively Z) be the intersection of DF (respectively DE) and the line through A parallel to BC. If E' and F' are the midpoints of DZ and DY, then the six points A, E, F, I, E', F' are on the same circle.

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, ∠EDC = ∠DEC (since CE and CD are two tangents to a circle from the same point) and ∠DEC = ∠AEZ (as vertical). We may conclude that ∠AEZ = ∠EZA so that ΔAEZ is isosceles: AE = AZ. Similarly, AF = AY. But AE and AF are the tangents from A to the incircle of ΔABC. Thus all four segments are equal:

AY = AF = AE = AZ.

In ΔEYZ, the median from E to YZ equals half of the latter, implying that the triangle is right: ∠YEZ = 90°. Similarly, ∠YFZ = 90°. This makes YE and ZF altitudes of ΔDYZ. Their intersection Ha is the orthocenter of that triangle. It follows that DHa is the third altitude and, since BC||YZ, DHa ⊥ BC. But D is the point of tangency of the incircle of ΔABC with BC, implying ID ⊥ BC and subsequently the collinearity of D, I, and Ha. Further, the right angle DEHa (with E and D on the incircle) needs to be subtended by a diameter, placing Ha on that circle and making Ha antipodal to D: DI = IHa.

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

  1. B. Suceava, P. Yiu, The Feuerbach Point and Euler lines, Forum Geometricorum, Volume 6 (2006) 191-197.

Nine Point Circle

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