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Cubics Passing Through Gergonne Point

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  • Quang Tuan Bui
    Dear All My Friends, Given triangle ABC with one point P with barycentrics (p : q : r) and any point X with barycentrics (x : y : z). Three lines La, Lb, Lc
    Message 1 of 1 , Jun 24, 2007
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      Dear All My Friends,
      Given triangle ABC with one point P with barycentrics (p : q : r) and any point X with barycentrics (x : y : z). Three lines La, Lb, Lc are passing through P and parallel with BC, CA, AB respectively.
      A' = intersection of AX and La
      B' = intersection of BX and Lb
      C' = intersection of CX and Lc
      The result: three segments PA', PB', PC' hold a property "one is sum of other two" if P is on the cubic with barycentric equation:

      CyclicSum[x*y*((x + y)*r*(a - b) + (q*x - p*y)*c + z*p*(b - c))] = 0

      This cubic is always passing through following points:
      - three vertices of ABC
      - three vertices of antimedial triangle
      - Gergonne point
      - Point P
      This cubic is self-isotomic iff P is on the Nagel line. In this case, cubic passes also through Centroid and Nagel point.
      This cubic could not be self-isogonal.
      This cubic intersect sidelines of ABC at Cevian traces of one point iff P is on Nagel line or P is on circumconic with perspector at X(9).

      Following are some special interesting cases:
      P = X(1) the cubic is Spieker perspector cubic, K034
      P = X(2) the cubic is Soddy-Gergonne-Nagel cubic, K200
      P = X(8) the cubic is Lucas cubic, K007
      P = X(78) the cubic is K154
      P = X(9) the cubic is one central cubic with center at X(9) and passing through following centers: X(7), X(9), X(144). It is interesting that K202 is also central cubic with center at X(9) and passing also through X(7), X(9), X(144). But K202 passes also through X(366). Our cubic does not.

      Some self-isotomic cubics:
      P = X(10) the cubic passes through: X(2), X(7), X(8), X(10), X(79), X(86), X(319), X(1029), X(2895)
      P= X(42) the cubic passes through: X(2), X(7), X(8), X(42), X(256), X(310), X(1655), X(1909)
      Some not self-isotomic cubics:
      P = X(21) the cubic passes through X(1), X(7), X(21), X(63), X(286), X(3219)
      P = X(81) the cubic passes through X(1), X(7), X(63), X(81), X(314), X(330), X(894)

      Please give me some advices about these facts!
      Are there any other interesting from these cubics?

      Thank you and best regards,
      Bui Quang Tuan



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