A Lemma?
- Thamim Zahin
- Posts:98
- Joined:Wed Aug 03, 2016 5:42 pm
In a $\triangle ABC$, let $H_A,H_B,H_C$ be the projection of $A,B,C$ on $BC,CA,AB$ respectively. And $O$ is the circumcenter
1. Prove that, $OA \perp H_BH_C$.
2. Let a line $\lambda$ be tangent to $(HBC)$ at $H$. Prove that, $OA \perp \lambda$.
1. Prove that, $OA \perp H_BH_C$.
2. Let a line $\lambda$ be tangent to $(HBC)$ at $H$. Prove that, $OA \perp \lambda$.
Last edited by Thamim Zahin on Sat Apr 15, 2017 1:24 pm, edited 3 times in total.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.
Re: A Lemma?
Define $H_BH_C$.
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.
- Charles Caleb Colton
- Charles Caleb Colton
- Raiyan Jamil
- Posts:138
- Joined:Fri Mar 29, 2013 3:49 pm
Re: A Lemma?
$H_B$ is the feet of perpendicular from $B$ to $AC$ nd figure $H_C$ urself.dshasan wrote:Define $H_BH_C$.
A smile is the best way to get through a tough situation, even if it's a fake smile.
- Thanic Nur Samin
- Posts:176
- Joined:Sun Dec 01, 2013 11:02 am
Re: A Lemma?
This note might be relevant.
Hammer with tact.
Because destroying everything mindlessly isn't cool enough.
Because destroying everything mindlessly isn't cool enough.
- Thanic Nur Samin
- Posts:176
- Joined:Sun Dec 01, 2013 11:02 am
Re: A Lemma?
Also, an elegant proof for part 2 is presented below.
Take the homothety with center $A$ and ratio $\dfrac{1}{2}$. This sends $(HBC)$ to the nine point circle. Let the center of the nine point circle be $N$, and let the midpoint of $AH$ be $M$. Now, since $N$ is the midpoint of $OH$, $MN||AO$[typo edited], and since $H$ is sent to $M$, the tangent is perpendicular to $MN$, which is consequently parallel to $AO$. Thus the original tangent was also perpendicular to $AO$.
Take the homothety with center $A$ and ratio $\dfrac{1}{2}$. This sends $(HBC)$ to the nine point circle. Let the center of the nine point circle be $N$, and let the midpoint of $AH$ be $M$. Now, since $N$ is the midpoint of $OH$, $MN||AO$[typo edited], and since $H$ is sent to $M$, the tangent is perpendicular to $MN$, which is consequently parallel to $AO$. Thus the original tangent was also perpendicular to $AO$.
Last edited by Thanic Nur Samin on Mon Apr 17, 2017 8:32 pm, edited 1 time in total.
Hammer with tact.
Because destroying everything mindlessly isn't cool enough.
Because destroying everything mindlessly isn't cool enough.
- Mallika Prova
- Posts:6
- Joined:Thu Dec 05, 2013 7:44 pm
- Location:Mymensingh,Bangladesh
Re: A Lemma?
a proof not elegant
1.
$\angle ABH_a =\angle AH_bH_c $ and
$\angle BAH_a =\angle H_bAO $ as $O$ and $H$ are isogonal conjugates
2.
definitely not elegant
let the tangent meet $AB$ and $BC$ at $E$,$F$
$\angle H_bHF=\angle EHB =\angle HCB=\angle HH_bH_c $
and gives $EF\|H_bH_c$
1.
$\angle ABH_a =\angle AH_bH_c $ and
$\angle BAH_a =\angle H_bAO $ as $O$ and $H$ are isogonal conjugates
2.
definitely not elegant
let the tangent meet $AB$ and $BC$ at $E$,$F$
$\angle H_bHF=\angle EHB =\angle HCB=\angle HH_bH_c $
and gives $EF\|H_bH_c$
Last edited by Mallika Prova on Mon Apr 17, 2017 8:28 pm, edited 1 time in total.
If you do what interests you , atleast one person is pleased.
- Mallika Prova
- Posts:6
- Joined:Thu Dec 05, 2013 7:44 pm
- Location:Mymensingh,Bangladesh
Re: A Lemma?
surely it wants to be $MN||AO$ ??Thanic Nur Samin wrote: $MN||AH$
If you do what interests you , atleast one person is pleased.
- ahmedittihad
- Posts:181
- Joined:Mon Mar 28, 2016 6:21 pm
Re: A Lemma?
I think you have the wrong idea of elegance. To see what not elegant is, Thanic Nur Samin vaiya is your best choice.
Frankly, my dear, I don't give a damn.
- Thanic Nur Samin
- Posts:176
- Joined:Sun Dec 01, 2013 11:02 am
Re: A Lemma?
Thanks I edited that.Mallika Prova wrote: surely it wants to be $MN||AO$ ??
Hammer with tact.
Because destroying everything mindlessly isn't cool enough.
Because destroying everything mindlessly isn't cool enough.