## Circumcenter and Incenter

Expand Messages
• Dear Friends Let ABC and O and I are its circumcenter and incenter,respectively. Let M be a point on the BC and U and V are the incenters of triangles ABM and
Message 1 of 4 , Apr 2, 2012
Dear Friends

Let ABC and O and I are its circumcenter and incenter,respectively.
Let M be a point on the BC and U and V are the incenters of triangles ABM and ACM. Let T the circumcenter of triangle AUV.
prove that OT and AI are parallel if and only if M = midpoint of BC

With Regards.
• Suppose, AN is the isogonal conjugate of AM wrt angle BAC where N lies on the circumcircle of ABC. External angle-bisector of angle BAC intersects BC at X.
Message 2 of 4 , Apr 2, 2012
Suppose, AN is the isogonal conjugate of AM wrt \angle BAC where N lies on the circumcircle of ABC.
External angle-bisector of \angle BAC intersects BC at X.
Under inversion wrt A followed by a reflection on the angle-bisector of BAC, suppose N goes to M.
Then U',V' be the images of U,V.
Clearly, U',V' are the A-excenters of ANC and ANB.
K=AU'\cap NC
L=AV'\cap NB
P,Q be the incenters of ANC and ANB. Note that, PQ,KL,U'V' are concurrent. Also KL intersects BC at X.
And from AI||OT we get that U'V' intersects BC at X too.
So PQ,KL,U'V',BC are concurrent.
AU'\cap BC=J, AV'\cap BC=H
So (AJ;KP)=(AH;LQ). So CP,BQ meet on AN.
So BN/NC=AB/AC. So AN is the A-symmedian of ABC.
So M is midpoint of BC.
• Dear Hyacinthists, Dear mbabelian,   it would be nice to mention the origin of this problem
Message 3 of 4 , Apr 2, 2012
Dear Hyacinthists,
Dear mbabelian,

it would be nice to mention the origin of this problem
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=472759
the author being Jean-Louis Ayme

Sincerely
Jean-Louis

________________________________
De : mbabelian <mbabelian@...>
À : Hyacinthos@yahoogroups.com
Envoyé le : Lundi 2 avril 2012 21h43
Objet : [EMHL] Circumcenter and Incenter

Dear Friends

Let ABC and O and I are its circumcenter and incenter,respectively.
Let M be a point on the BC and U and V are the incenters of triangles ABM and ACM. Let T the circumcenter of triangle AUV.
prove that OT and AI are parallel if and only if M = midpoint of BC

With Regards.

[Non-text portions of this message have been removed]
• Dear Friends, Dear Jean-Louis I just found it in a weblog and it didn t mention the author of it. sorry. Thank you for mentioning its origin. With Regards.
Message 4 of 4 , Apr 2, 2012
Dear Friends,
Dear Jean-Louis
I just found it in a weblog and it didn't mention the author of it. sorry.
Thank you for mentioning its origin.
With Regards.

--- In Hyacinthos@yahoogroups.com, Jean-Louis Ayme <jeanlouisayme@...> wrote:
>
> Dear Hyacinthists,
> Dear mbabelian,
> Â
> it would be nice to mention the origin of this problem
> http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=472759
> the author being Jean-Louis Ayme
> Â
> Sincerely
> Jean-Louis
>
>
> ________________________________
> DeÂ : mbabelian <mbabelian@...>
> ÃÂ : Hyacinthos@yahoogroups.com
> EnvoyÃ© le : Lundi 2 avril 2012 21h43
> ObjetÂ : [EMHL] Circumcenter and Incenter
>
>
> Â
> Dear Friends
>
> Let ABC and O and I are its circumcenter and incenter,respectively.
> Let M be a point on the BC and U and V are the incenters of triangles ABM and ACM. Let T the circumcenter of triangle AUV.
> prove that OT and AI are parallel if and only if M = midpoint of BC
>
> With Regards.
>
>
>
>
> [Non-text portions of this message have been removed]
>
Your message has been successfully submitted and would be delivered to recipients shortly.