SEQUENCE OF POINTS ON THE EULER LINE

Let ABC be a triangle.

Denote:

A1, B1, C1 = The NPC centers of NBC, NCA, NAB, resp.

A2, B2 ,C2 = The NPC centers of A1BC, B1CA, C1AB, resp.

A3, B3, C3 = The NPC centers of A2BC, B2CA, C2AB

An, Bn, Cn = The NPC centers of A_n-1BC, B_n-1CA, C_n-1AB.

On = the circumcenter of the triangle AnBnCn.

The points O1, O2,......, On, ..... lie on the Euler Line of ABC.

The point O1 is now in ETC: X5501

Coordinates of On? Ratio of OnO / OnN ?

Antreas P. Hatzipolakis, 14 April 2013

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I calculated the ratios NO1:O1O and NO2:O2O. The expression for the first one is quite long, and that for the second one is enormous:

If p stands for the semiperimeter, we have

NO1:O1O = (2 p^4 - 12 p^2 r^2 + 2 r^4 - 16 p^2 r R + 16 r^3 R - 8 p^2 R^2 + 40 r^2 R^2 + 32 r R^3 + 9 R^4)/(2 p^4 + 20 p^2 r^2 + 2 r^4 - 16 p^2 r R + 16 r^3 R - 16 p^2 R^2 + 48 r^2 R^2 + 64 r R^3 + 23 R^4)

Francisco Javier, Hyacinthos #21952

The conjecture is false !!

From Cesar Lozada (24 June 2016)

Dear Antreas,

Sorry. O1 does lie on Euler line but O2 does not. Algebraically confirmed.

I didn´t try O3 because calculus are awful

Regards,

César Lozada