Isogonal conjugate of nine points circle wrt its triangle

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• Dear all (1).\$ABC\$ a triangle and \$I\$ is its Incenter. (2)Incircle of \$ABC\$ meet \$BC,CA,AB\$ at \$D,E,F\$ respectively. (3)\$D\$ is the midpoint of \$IP\$. (4)\$N \$ is
Message 1 of 3 , Jul 9, 2012
Dear all

(1).\$ABC\$ a triangle and \$I\$ is its Incenter.
(2)Incircle of \$ABC\$ meet \$BC,CA,AB\$ at \$D,E,F\$ respectively.
(3)\$D\$ is the midpoint of \$IP\$.
(4)\$N'\$ is the Isogonal conjugate of \$N\$,the center of nine points circle of \$DEF\$ wrt triangle \$DEF\$.

prove that \$N',A,P\$ lie on a line.

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are there any article or paper about N's Isogonal conjugate?
-------------------------------------------

With Regards.
• Your point N is X(3649) = KS(INTOUCH TRIANGLE) However, N , A and P are never collinear, except when ABC is isosceles with AB=AC. Best regards, Francisco
Message 2 of 3 , Jul 9, 2012
Your point N' is X(3649) = KS(INTOUCH TRIANGLE)

However, N', A and P are never collinear, except when ABC is isosceles with AB=AC.

Best regards,

Francisco Javier.

--- In Hyacinthos@yahoogroups.com, "mbabelian" <mbabelian@...> wrote:
>
> Dear all
>
> (1).\$ABC\$ a triangle and \$I\$ is its Incenter.
> (2)Incircle of \$ABC\$ meet \$BC,CA,AB\$ at \$D,E,F\$ respectively.
> (3)\$D\$ is the midpoint of \$IP\$.
> (4)\$N'\$ is the Isogonal conjugate of \$N\$,the center of nine points circle of \$DEF\$ wrt triangle \$DEF\$.
>
> prove that \$N',A,P\$ lie on a line.
>
> -------------------------------------------
> are there any article or paper about N's Isogonal conjugate?
> -------------------------------------------
>
> With Regards.
>
• As Chandan Banerjee have just pointed out from Argentina, this is true: reflection of I on N is collinear with A,P. Good luck at IMO!
Message 3 of 3 , Jul 9, 2012
As Chandan Banerjee have just pointed out from Argentina, this is true: reflection of I on N' is collinear with A,P.

Good luck at IMO!

--- In Hyacinthos@yahoogroups.com, "mbabelian" <mbabelian@...> wrote:
>
> Dear all
>
> (1).\$ABC\$ a triangle and \$I\$ is its Incenter.
> (2)Incircle of \$ABC\$ meet \$BC,CA,AB\$ at \$D,E,F\$ respectively.
> (3)\$D\$ is the midpoint of \$IP\$.
> (4)\$N'\$ is the Isogonal conjugate of \$N\$,the center of nine points circle of \$DEF\$ wrt triangle \$DEF\$.
>
> prove that \$N',A,P\$ lie on a line.
>
> -------------------------------------------
> are there any article or paper about N's Isogonal conjugate?
> -------------------------------------------
>
> With Regards.
>
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