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22174Circles, Circles, Circles....... (was: Envelopes)

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  • Antreas Hatzipolakis
    Apr 23, 2014
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    Let X,Y,Z be three lines.

    Denote:
    xL1, xL2, xL3 = the parallels to X through A,B,C, resp.
    yL1, yL2, yL3 = the parallels to Y through A,B,C, resp.
    zL1, zL2, zL3 = the parallels to Z through A,B,C, resp.

    xAxBxC = the triangle bounded by the reflections of xL1, xL2, xL3 in BC,CA,AB, resp.
    yAyByC = the triangle bounded by the reflections of yL1, yL2, yL3 in BC,CA,AB, resp.
    zAzBzC = the triangle bounded by the reflections of zL1, zL2, zL3 in BC,CA,AB, resp.

    eX, eY, eZ = the Euler lines of xAxBxC, yAyByC, zAzBzC, resp.

    rX, rY, rZ = the radical axis of (O) and (N) of the triangles xAxBxC, yAyByC, zAzBzC, resp.

    We have seen that xE, yE, zE pass through a fixed point, and also xR, yR, zR pass through an other fixed point.

    Now,
    xeX = the parallel through A to the line Euler line eX
    yeY = the parallel through B to the line eY
    zeZ = the parallel through C to the line eZ

    xrX = the parallel through A to the line (radical axis) rX
    yrY = the parallel through B to the line rY
    zrZ = the parallel through C to the line rZ

    Question: For which lines X,Y,Z the lines xeX, yeY, zeZ and also the lines
    xrX, yrY, zrZ are concurrent?

    Some examples of loci:
    Two of the lines X,Y,Z are fixed and the third is variable (trilinear polar of a variable
    point P). Or we can take X,Y,Z = cevians of a variable point P or sidelines
    of a variable triangle (pedal or cevian triangle of a point P).

    Two special cases of lines X.Y,Z:

    The lines X,Y,Z are:
    1. the sidelines BC,CA,AB, resp.
    2. the altitude lines a_altitude, b_altitude, c_altitude, resp..

    [in the first case we have X = BC = xL2 = xL3  etc and in the second  X = xL1 etc]

    Observations:

    For case 1. The lines xeX, yeY, zeZ concur on a point Q1 on the circumcircle of ABC
    and the lines xrX, yrY, zrZ on another point Q2 on the circumcircle.
    For case 2. The lines xeX, yeY, zeZ concur on a point T1 on the circumcircle of ABC
    and the lines xrX, yrY, zrZ on an other  point T2 on the circumcircle.
    We have Q1 = T2 and Q2 = T1

    Let 1Oa, 1Ob,1Oc be the circumcenters of xAxBxC,yAyByC, zAzBzC, resp.  for case 1, and 2Oa,2Ob,2Oc for case 2.
    The six circumcenters are concyclic. The same is true for the NPC centers of the triangles.

    In general:
    Let 1Pa, 1Pb, 1Pc be three points on the Euler lines eX, eY, eZ of the case 1 and
    2Pa,2Pb, 2Pc three points on the Euler lines eX, eY, eZ of the case 2, such that all six points are same
    (ie have same coordinates with respect to six triangles ie they are all circumcenters, orthocenters, NPC centers etc)
    The six points are concyclic. The circumcenters of the circles, as P's move on the Euler lines, being same,
    are collinear on a line k, and have same positions on the line k as the P's on the Euler lines.
    [this means, for exampe, that if O',N',H' are the circumcenters on the line k corresponding to Os, Ns, H's on the Euler lines,
    then |N'O'| = |N'H'| as |NO| = |NH|)

    Note: How about if the lines of the cases 1 and 2 are perpendiculars in general but not necessarily sidelines and altitudes?
    We have again concyclic points?

    Hope there are no too many errors, typos etc .... :-)

    Antreas


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