HYACINTHOS 23764

[Seiichi Kirikami]:

 

 

Dear friends,

 

Let ABC be a triangle and Ia, Ib, Ic its excenters.

Denote:

Ja, Jb, Jc = the incenters of IaBC, IbCA, IcAB.

l, n, m = the Euler lines of IaJbJc, JaIbJc, IaIbIc.

l, m, n concur in a point P.

Search number of triangle {6, 9, 13} =5.0703580242..

I do not have the coordinates of P yet.

See the attachment.

 

Best regards, Seiichi.

 

[APH]:

Dear Seiichi,

Inspired from your construction:

Let ABC be a triangle and OaObOc the antipedal triangle of O (tangential triangle).

Denote:

Ha, Hb, Hc = the orthocenters of OaBC, ObCA, OcAB, resp.

The Euler lines of OaHbHc, ObHcHa, OcHaHb and ABC

are concurrent.

Antreas P. Hatzipolakis

 

[Seiichi Kirikami]:

Dear Antreas,

 

1st barycentric coordinate of your problem (the antipedal triangle of O) =

-a^8 (b^2 + c^2) + 2 a^4 b^2 c^2 (b^2 + c^2) + (b^2 - c^2)^4 (b^2 + c^2) -

2 a^2 (b^2 - c^2)^2 (b^4 + c^4) + 2 a^6 (b^4 - b^2 c^2 + c^4).

Best regard, Seiichi Kirikami