*#424************************
Dear Members,
I have just discovered following problem:
Let P be a point in the plane of a triangle ABC,
lines AP, BP and CP intersect sides BC, CA and AB at points D, E and F,
respectively.
Points Ha, Hb and Hc are orthocenters of triangles AEF, BDF and CDE
respectively. Then points, Ha, Hb, Hc, D, E, F lie on a conic.
Best regards,
Dominik Burek
*#425************************
Dear Dominik,
A special case of this occurs if P lies on the Lucas cubic. Then DEF and HaHbHc are homothetic (at the conic center), and HaHbHc is perspective to ABC at a point on the Darboux cubic. I did not see these properties listed on Bernard's site.
Some specifics for P on the Lucas cubic, Q the conic center, and Y the perspector of HaHbHc and ABC:
P=X(2), Q=X(5), Y=X(4)
P=X(4), Q=X(389), Y=X(3)
P=X(7), Q=X(942), Y=X(1)
P=X(8), Q=(non-ETC, -0.004273928211234)
= midpoint of X(4) and X(72)
= complement of X(1071)
= X(5) of triangle formed by lines through external pairs of extouch points (the '2nd extouch triangle')
Y=X(84)
P=X(20), Q=X(3), Y=X(3346)
P=X(69), Q=(non-ETC, -1.913504596254703)
= complement of X(185)
= (see below)
Y=X(64)
P=X(189), Q=(non-ETC, 16.749225183870210), Y=X(40)
P=X(253), Q=X(4), Y=X(20)
P=X(329), Q=(non-ETC, -15.643481700600276), Y=X(3345)
P=X(1032), Q=(non-ETC, 0.287119307452631), Y=X(1498)
P=X(1034), Q=(non-ETC, -0.838248707963038), Y=X(1490)
Another property of Q for P=X(69):
Let A'B'C' be the half-altitude triangle.
Let A" be the trilinear pole, wrt A'B'C', of line BC, and define B", C" cyclically.
Let A* be the trilinear pole, wrt A'B'C', of line B"C", and define B*, C* cyclically.
The lines A'A*, B'B*, C'C* concur in Q=complement of X(185).
Question: what is the locus of Q for P on the Lucas cubic (includes X(3), X(4), X(5), X(389), X(942), and the above non-ETC points)?
Best regards,
Randy Hutson
*#431************************
Deas Randy
a central cubic with center X(5) and three real asymptotes joining X(5) and the midpoints of ABC.
I didn't find other ETC centers on the curve.
Best regards
Bernard Gibert
barycentric equation :
a^10*b^2*x^3 - 5*a^8*b^4*x^3 + 10*a^6*b^6*x^3 - 10*a^4*b^8*x^3 +
5*a^2*b^10*x^3 - b^12*x^3 - a^10*c^2*x^3 + 2*a^6*b^4*c^2*x^3 +
4*a^4*b^6*c^2*x^3 - 9*a^2*b^8*c^2*x^3 + 4*b^10*c^2*x^3 +
5*a^8*c^4*x^3 - 2*a^6*b^2*c^4*x^3 + 2*a^2*b^6*c^4*x^3 - 5*b^8*c^4*x^3 -
10*a^6*c^6*x^3 - 4*a^4*b^2*c^6*x^3 - 2*a^2*b^4*c^6*x^3 + 10*a^4*c^8*x^3 +
9*a^2*b^2*c^8*x^3 + 5*b^4*c^8*x^3 -
5*a^2*c^10*x^3 - 4*b^2*c^10*x^3 + c^12*x^3 + a^12*x^2*y -
3*a^10*b^2*x^2*y + 10*a^6*b^6*x^2*y - 15*a^4*b^8*x^2*y + 9*a^2*b^10*x^2*y -
2*b^12*x^2*y + 2*a^10*c^2*x^2*y +
a^8*b^2*c^2*x^2*y - 16*a^6*b^4*c^2*x^2*y + 22*a^4*b^6*c^2*x^2*y -
10*a^2*b^8*c^2*x^2*y + b^10*c^2*x^2*y - 17*a^8*c^4*x^2*y +
10*a^6*b^2*c^4*x^2*y + 16*a^4*b^4*c^4*x^2*y -
10*a^2*b^6*c^4*x^2*y + b^8*c^4*x^2*y + 28*a^6*c^6*x^2*y -
6*a^4*b^2*c^6*x^2*y + 16*a^2*b^4*c^6*x^2*y + 10*b^6*c^6*x^2*y -
17*a^4*c^8*x^2*y - 7*a^2*b^2*c^8*x^2*y - 16*b^4*c^8*x^2*y +
2*a^2*c^10*x^2*y + 5*b^2*c^10*x^2*y + c^12*x^2*y + 2*a^12*x*y^2 -
9*a^10*b^2*x*y^2 + 15*a^8*b^4*x*y^2 - 10*a^6*b^6*x*y^2 + 3*a^2*b^10*x*y^2 -
b^12*x*y^2 - a^10*c^2*x*y^2 +
10*a^8*b^2*c^2*x*y^2 - 22*a^6*b^4*c^2*x*y^2 + 16*a^4*b^6*c^2*x*y^2 -
a^2*b^8*c^2*x*y^2 - 2*b^10*c^2*x*y^2 - a^8*c^4*x*y^2 +
10*a^6*b^2*c^4*x*y^2 - 16*a^4*b^4*c^4*x*y^2 -
10*a^2*b^6*c^4*x*y^2 + 17*b^8*c^4*x*y^2 - 10*a^6*c^6*x*y^2 -
16*a^4*b^2*c^6*x*y^2 + 6*a^2*b^4*c^6*x*y^2 - 28*b^6*c^6*x*y^2 +
16*a^4*c^8*x*y^2 + 7*a^2*b^2*c^8*x*y^2 +
17*b^4*c^8*x*y^2 - 5*a^2*c^10*x*y^2 - 2*b^2*c^10*x*y^2 - c^12*x*y^2 +
a^12*y^3 - 5*a^10*b^2*y^3 + 10*a^8*b^4*y^3 - 10*a^6*b^6*y^3 +
5*a^4*b^8*y^3 - a^2*b^10*y^3 - 4*a^10*c^2*y^3 +
9*a^8*b^2*c^2*y^3 - 4*a^6*b^4*c^2*y^3 - 2*a^4*b^6*c^2*y^3 +
b^10*c^2*y^3 + 5*a^8*c^4*y^3 - 2*a^6*b^2*c^4*y^3 + 2*a^2*b^6*c^4*y^3 -
5*b^8*c^4*y^3 + 2*a^4*b^2*c^6*y^3 +
4*a^2*b^4*c^6*y^3 + 10*b^6*c^6*y^3 - 5*a^4*c^8*y^3 - 9*a^2*b^2*c^8*y^3 -
10*b^4*c^8*y^3 + 4*a^2*c^10*y^3 + 5*b^2*c^10*y^3 - c^12*y^3 - a^12*x^2*z -
2*a^10*b^2*x^2*z +
17*a^8*b^4*x^2*z - 28*a^6*b^6*x^2*z + 17*a^4*b^8*x^2*z -
2*a^2*b^10*x^2*z - b^12*x^2*z + 3*a^10*c^2*x^2*z - a^8*b^2*c^2*x^2*z -
10*a^6*b^4*c^2*x^2*z + 6*a^4*b^6*c^2*x^2*z +
7*a^2*b^8*c^2*x^2*z - 5*b^10*c^2*x^2*z + 16*a^6*b^2*c^4*x^2*z -
16*a^4*b^4*c^4*x^2*z - 16*a^2*b^6*c^4*x^2*z + 16*b^8*c^4*x^2*z -
10*a^6*c^6*x^2*z - 22*a^4*b^2*c^6*x^2*z +
10*a^2*b^4*c^6*x^2*z - 10*b^6*c^6*x^2*z + 15*a^4*c^8*x^2*z +
10*a^2*b^2*c^8*x^2*z - b^4*c^8*x^2*z - 9*a^2*c^10*x^2*z - b^2*c^10*x^2*z +
2*c^12*x^2*z + a^12*y^2*z +
2*a^10*b^2*y^2*z - 17*a^8*b^4*y^2*z + 28*a^6*b^6*y^2*z -
17*a^4*b^8*y^2*z + 2*a^2*b^10*y^2*z + b^12*y^2*z + 5*a^10*c^2*y^2*z -
7*a^8*b^2*c^2*y^2*z - 6*a^6*b^4*c^2*y^2*z +
10*a^4*b^6*c^2*y^2*z + a^2*b^8*c^2*y^2*z - 3*b^10*c^2*y^2*z -
16*a^8*c^4*y^2*z + 16*a^6*b^2*c^4*y^2*z + 16*a^4*b^4*c^4*y^2*z -
16*a^2*b^6*c^4*y^2*z + 10*a^6*c^6*y^2*z -
10*a^4*b^2*c^6*y^2*z + 22*a^2*b^4*c^6*y^2*z + 10*b^6*c^6*y^2*z +
a^4*c^8*y^2*z - 10*a^2*b^2*c^8*y^2*z - 15*b^4*c^8*y^2*z + a^2*c^10*y^2*z +
9*b^2*c^10*y^2*z - 2*c^12*y^2*z -
2*a^12*x*z^2 + a^10*b^2*x*z^2 + a^8*b^4*x*z^2 + 10*a^6*b^6*x*z^2 -
16*a^4*b^8*x*z^2 + 5*a^2*b^10*x*z^2 + b^12*x*z^2 + 9*a^10*c^2*x*z^2 -
10*a^8*b^2*c^2*x*z^2 -
10*a^6*b^4*c^2*x*z^2 + 16*a^4*b^6*c^2*x*z^2 - 7*a^2*b^8*c^2*x*z^2 +
2*b^10*c^2*x*z^2 - 15*a^8*c^4*x*z^2 + 22*a^6*b^2*c^4*x*z^2 +
16*a^4*b^4*c^4*x*z^2 - 6*a^2*b^6*c^4*x*z^2 -
17*b^8*c^4*x*z^2 + 10*a^6*c^6*x*z^2 - 16*a^4*b^2*c^6*x*z^2 +
10*a^2*b^4*c^6*x*z^2 + 28*b^6*c^6*x*z^2 + a^2*b^2*c^8*x*z^2 -
17*b^4*c^8*x*z^2 - 3*a^2*c^10*x*z^2 + 2*b^2*c^10*x*z^2 +
c^12*x*z^2 - a^12*y*z^2 - 5*a^10*b^2*y*z^2 + 16*a^8*b^4*y*z^2 -
10*a^6*b^6*y*z^2 - a^4*b^8*y*z^2 - a^2*b^10*y*z^2 + 2*b^12*y*z^2 -
2*a^10*c^2*y*z^2 + 7*a^8*b^2*c^2*y*z^2 -
16*a^6*b^4*c^2*y*z^2 + 10*a^4*b^6*c^2*y*z^2 + 10*a^2*b^8*c^2*y*z^2 -
9*b^10*c^2*y*z^2 + 17*a^8*c^4*y*z^2 + 6*a^6*b^2*c^4*y*z^2 -
16*a^4*b^4*c^4*y*z^2 - 22*a^2*b^6*c^4*y*z^2 +
15*b^8*c^4*y*z^2 - 28*a^6*c^6*y*z^2 - 10*a^4*b^2*c^6*y*z^2 +
16*a^2*b^4*c^6*y*z^2 - 10*b^6*c^6*y*z^2 + 17*a^4*c^8*y*z^2 -
a^2*b^2*c^8*y*z^2 - 2*a^2*c^10*y*z^2 + 3*b^2*c^10*y*z^2 -
c^12*y*z^2 - a^12*z^3 + 4*a^10*b^2*z^3 - 5*a^8*b^4*z^3 + 5*a^4*b^8*z^3 -
4*a^2*b^10*z^3 + b^12*z^3 + 5*a^10*c^2*z^3 - 9*a^8*b^2*c^2*z^3 +
2*a^6*b^4*c^2*z^3 - 2*a^4*b^6*c^2*z^3 +
9*a^2*b^8*c^2*z^3 - 5*b^10*c^2*z^3 - 10*a^8*c^4*z^3 + 4*a^6*b^2*c^4*z^3 -
4*a^2*b^6*c^4*z^3 + 10*b^8*c^4*z^3 + 10*a^6*c^6*z^3 + 2*a^4*b^2*c^6*z^3 -
2*a^2*b^4*c^6*z^3 -
10*b^6*c^6*z^3 - 5*a^4*c^8*z^3 + 5*b^4*c^8*z^3 + a^2*c^10*z^3 -
b^2*c^10*z^3=0
*#433************************
Dear Bernard,
Thank you! I did not expect the equation to be so complicated, given
the relatively simple equations for the Lucas and Darboux cubics.
It turns out that the antipode (reflection in X(5)) of X(389) in the
cubic is the center for P=X(69) (complement of X(185)), and the antipode
of X(942) is the center for P=X(8) (complement of X(1071)). In general,
it looks like if P1 and P2 are isotomic conjugates on the Lucas cubic,
then the respective conic centers, Q1 and Q2, are antipodes on the
central cubic with center X(5).
Best regards,
Randy Hutson
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