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

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  • Antreas Hatzipolakis
    Nov 6 4:02 AM
    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