ADGEOM 4554

[archimedes_26 = Kadir Altintas]:

Dear geometers

Let ABC be triangle and P any point.

Let O1,O2,O3 circumcenters of circles BCP, CAP and ABP.

Let Ha be hyperbola whose foci are O2 and O3 and passes through A.

Define Hb,Hc similarly.

These hyperbolas have always a second intersection point other than P. 

Is it possibelto find (in barycentrics) hyperbolic images of notable points ? (P=X(1), X(2).....)

What is the locus of points  if P moves on Euler line?

Thank you

With my regards....

 

[Randy Hutson]:

The hyperbolic image of P is the hyperbolic conjugate of P (discussed here sometime earlier) wrt O1O2O3.  Bernard Gibert has found that the coordinates for hyperbolic conjugates are in general very complicated, so I would expect the same for hyperbolic images.  One exception I found is when P = X(1), the hyperbolic image lies on lines {1,164}, {40,[X(108) of excentral triangle]}, {168,3973}, {361,1743}, {8688,12518} and has trilinears:

Cos[A/2] (Csc[B/2] - Csc[C/2]) - Cos[B/2] (Csc[C/2] - Csc[A/2]) - Cos[C/2] (Csc[A/2] - Csc[B/2]) - 2 Cot[B/2] (1 - Csc[C/2] Sin[A/2]) + 2 Cot[C/2] (1 - Csc[B/2] Sin[A/2]) : :

(6,9,13) values: (13.169691395932970, 9.337732849541849, -8.902238904359810)

Note that in this case, O1O2O3 is the 2nd circumperp triangle.


For P = X(2), the hyperbolic image is collinear with X(2) and [X(1) of circummedial triangle], but I could find no other lines through it.


For P = X(4), the hyperbolas are degenerate (the altitudes of ABC), and meet only at X(4).  O1O2O3 is the Johnson triangle in this case.


Best regards,
Randy Hutson