## 22012X(5569) Dao's Circle Center - Generalization

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• Nov 6 4:02 AM
• 179 KB
Let ABC be a triangle, and let Ab be the center of the circle through A and
tangent to the B-median, and define Bc and Ca cyclically. Let Ac be the center
of the circle through A and tangent to the C-median, and define Ba and Cb cyclically.
The points Ab, Ba, Bc, Cb, Ca, Ac lie on a circle, of which X(5569) is the center,
(Dao Thanh Oai, Nov. 3, 2013)

Equivalently:

Let ABC be a triangle, and let Ab be the intersection of the perpendicular
bisector of AG and the perpendicular to B-median at G, and define Bc, Ca
cyclically. Let Ac be the intersection of the perpendicular bisector of AG
and the perpendicular to C-median at G, and define Ba and Cb cyclically.
The points Ab, Ba, Bc, Cb, Ca, Ac lie on a circle, of which X(5569) is the center,

The circles (Ab), (Ac), (Bc), (Ba), (Ca), (Cb) concur at G

Generalization

Let ABC be a triangle and Gat, Gbt, Gct three points on AG,BC,CG, resp.
such that:

GatA / GatG = GbtB / GbtG = GctC / GctG. = t

Let Ab be the intersection of the perpendicular to  AG at Gat and the
perpendicular to B-median at G. and define Bc, Ca
cyclically. Let Ac be the
intersection of the perpendicular to AG at Gat
and the perpendicular to C-median
at G, and define Ba and Cb cyclically.
The points Ab, Ba, Bc, Cb, Ca, Ac lie on
a circle.

Locus: Which is the center of the circle as t varies?

APH

In the attached figure t = 1/3