## Cubics Passing Through Gergonne Point

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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
Message 1 of 1 , Jun 24, 2007
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)