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## Re: [EMHL] Re: A CONJECTURE on NEUBERG (was: Thomson ?)

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• Dear Jean-Pierre, ... So, we have the Theorem: Let AA , BB , CC be the three altitudes of ABC, and Let Ab, Ac be the orth. proj. of A on AB, AC resp. Bc, Ba
Message 1 of 2 , Sep 1, 2001
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Dear Jean-Pierre,

[APH]:
>> Let ABC be a triangle, and A'B'C' the pedal triangle of P.
>>
>>
>> A
>> /\
>> / \
>> / \
>> C' B'
>> / \
>> / P \
>> Ab Ac
>> / \
>> B-------A'-------C
>>
>> Let Ab, Ac be the orth. proj. of A' on AB, AC resp.
>> Bc, Ba " B' BC, BA
>> Ca, Cb " C' CA, CB
>> [...] is it true that the OH/OK lines of the triangles
>> A'AbAc, B'BcBa, C'CaCb concur [at the OH/OK line of A'B'C']
>> for P in {H, O, I} ?

[JPE]:
>You're right for P = H and the Eulerline : the four Euler lines
>concur.

So, we have the Theorem:

Let AA', BB', CC' be the three altitudes of ABC, and
Let Ab, Ac be the orth. proj. of A' on AB, AC resp.
Bc, Ba " B' BC, BA
Ca, Cb " C' CA, CB

Then the Euler lines of A'B'C', A'AbAc, B'BcBa, C'CaCb are concurrent.

Which is the point of concurrence [a point lying on the Euler line of
the orthic triangle A'B'C' of ABC]? Is it in ETC?

And another conjecture:
The Euler lines of the triangles AAbAc, BBcBa, CCaCb are concurrent.
[are also concurrent for P = I or O ?]

Thank You!

Antreas
• Dear Antreas and other Hyacinthists, ... concurrent. ... of ... X-442 of the orthic triangle (not in ETC, I think) ... Very good. Yes, they are, but not on the
Message 2 of 2 , Sep 1, 2001
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Dear Antreas and other Hyacinthists,
> [APH]:

> So, we have the Theorem:
>
> Let AA', BB', CC' be the three altitudes of ABC, and
> Let Ab, Ac be the orth. proj. of A' on AB, AC resp.
> Bc, Ba " B' BC, BA
> Ca, Cb " C' CA, CB
>
> Then the Euler lines of A'B'C', A'AbAc, B'BcBa, C'CaCb are
concurrent.
>
> Which is the point of concurrence [a point lying on the Euler line
of
> the orthic triangle A'B'C' of ABC]? Is it in ETC?

X-442 of the orthic triangle (not in ETC, I think)

> And another conjecture:
> The Euler lines of the triangles AAbAc, BBcBa, CCaCb are concurrent.

Very good. Yes, they are, but not on the Euler line of A'B'C'. I
cannot recognize the common point.

The locus of P for which The Euler lines of the triangles AAbAc,
BBcBa, CCaCb are concurrent is again a quartic.

> Are also concurrent for I and O ?

I don't think so.

Friendly. Jean-Pierre
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