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Re: [EMHL] Three collinear centers

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  • Barry Wolk
    BW ...    These 3 circles meet at the Feuerbach point and at ( (b-c)^4 (b+c-a)^3, ... , ...)   Does this work if the Feuerbach point is replaced with some
    Message 1 of 37 , Jan 18, 2013
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      BW
      > A possible generalization led to something curious. Take all circles that are
      > tangent to two of the sidelines BC, CA, AB.  Four of these circles pass through
      > the Feuerbach point, the incircle and 3 others. The centers of those 3 others
      > are collinear.
        

      These 3 circles meet at the Feuerbach point and at ( (b-c)^4 (b+c-a)^3, ... , ...)
       > Does this work if the Feuerbach point is replaced with some other point on the
      > incircle?

      I've solved this. If the Feuerbach point is changed to one of the 3 points where the incircle meets the sidelines, these are degenerate cases where there aren't 4 such circles. And the Feuerbach point is the only other point on the incircle where this collinearity holds.
      --
      Barry Wolk
    • Antreas
      It is Another Seven Circles Theorem ! The Seven Circles Theorem states: Let a,b ,c, a ,b,c be a closed chain of six circles, all toouching a base circle
      Message 37 of 37 , Mar 21 2:58 PM
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        It is Another "Seven Circles Theorem" !

        The "Seven Circles Theorem" states:

        Let a,b',c, a',b,c' be a closed chain of six circles,
        all toouching a base circle w[omega], and suppose that
        their points of contact with w are six distinct points
        A,B',C,A',B,C' respectively. Then, subject of a certain
        extra condition to be discussed below, AA',BB',CC' are concurrent.
        (C J A Evelyn, B G Money - Coutts, J A Tyrrell:
        The Seven Circles Theorem and other theorems. London 1974, p. 31)

        See also
        http://en.wikipedia.org/wiki/Seven_circles_theorem
        http://mathworld.wolfram.com/SevenCirclesTheorem.html

        APH


        --- In Hyacinthos@yahoogroups.com, "Angel" <amontes1949@...> wrote:
        >
        > Dear Antreas
        >
        > --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
        > >
        > > Dear Alex
        > >
        > > Angel has tested the loci problems and it seems that
        > > for all P's the lines are concurrent for the cevian case,
        > > and for the pedal case as well except for the points P on the
        > > bisectors and on the circumcircle.
        > >
        > > Hmmmmmmm...... Maybe it is true in general ie for any
        > > circle intersecting the sidelines of the triangle!!
        > >
        >
        >
        > A check with GeoGebra:
        >
        >
        >
        > Let ABC be a triangle and P a point.
        >
        > Let (Q) be the CEVIAN circle of P and (Qab), (Qac) the circles touching AB,AC and (Q) internally and let (Tab), (Tac) be the points of contact. Similarly we define the points Tbc, Tba and Tca, Tcb.
        >
        > For all P (except for the points P on the bisectors) the lines TabTac, TbcTba, TcaTcb are concurrent.
        >
        > http://amontes.webs.ull.es/geogebra/Hyacinthos21421.html
        >
        > ----------------------
        >
        > Let (Q) be the PEDAL circle of P and (Qab), (Qac) the circles touching AB,AC and (Q) internally and let (Tab), (Tac) be the points of contact. Similarly we define the points Tbc, Tba and Tca, Tcb.
        >
        > For all P (except for the points P on the
        > bisectors and on the circumcircle) the lines TabTac, TbcTba, TcaTcb are concurrent.
        >
        > http://amontes.webs.ull.es/geogebra/Hyacinthos21421Pedal.html
        >
        > ------------------------------------------------
        >
        > Let (Q) be the circle Intersecting the sidelines of the triangle in D, E and F. (Qab), (Qac) the circles touching AB,AC and (Q) internally and let (Tab), (Tac) be the points of contact. Similarly we define the
        > points Tbc, Tba and Tca, Tcb.
        >
        > The lines TabTac, TbcTba, TcaTcb are concurrent.
        >
        > http://amontes.webs.ull.es/geogebra/Hyacinthos21428.html
        >
        > Best regards,
        > Angel M.
        >
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