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BdMO Online Forum • View topic - A Lemma?

A Lemma?

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A Lemma?

Post Number:#1  Unread postby Thamim Zahin » Fri Apr 14, 2017 6:09 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, $OA \perp \lambda$.
Last edited by Thamim Zahin on Sat Apr 15, 2017 1:24 pm, edited 3 times in total.
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Re: A Lemma?

Post Number:#2  Unread postby dshasan » Fri Apr 14, 2017 10:41 pm

Define $H_BH_C$.
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Re: A Lemma?

Post Number:#3  Unread postby Raiyan Jamil » Fri Apr 14, 2017 10:51 pm

dshasan wrote:Define $H_BH_C$.

$H_B$ is the feet of perpendicular from $B$ to $AC$ nd figure $H_C$ urself.
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Re: A Lemma?

Post Number:#4  Unread postby Thanic Nur Samin » Mon Apr 17, 2017 12:27 pm

Hammer with tact.

Because destroying everything mindlessly isn't cool enough.
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Re: A Lemma?

Post Number:#5  Unread postby Thanic Nur Samin » Mon Apr 17, 2017 12:40 pm

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 $AO$.
Last edited by Thanic Nur Samin on Mon Apr 17, 2017 8:32 pm, edited 1 time in total.
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Re: A Lemma?

Post Number:#6  Unread postby Mallika Prova » Mon Apr 17, 2017 7:52 pm

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 $EF\|H_bH_c$
Last edited by Mallika Prova on Mon Apr 17, 2017 8:28 pm, edited 1 time in total.
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Re: A Lemma?

Post Number:#7  Unread postby Mallika Prova » Mon Apr 17, 2017 8:01 pm

Thanic Nur Samin wrote: $MN||AH$


surely it wants to be $MN||AO$ ??
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Re: A Lemma?

Post Number:#8  Unread postby ahmedittihad » Mon Apr 17, 2017 8:03 pm

I think you have the wrong idea of elegance. To see what not elegant is, Thanic Nur Samin vaiya is your best choice.
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Re: A Lemma?

Post Number:#9  Unread postby Thanic Nur Samin » Mon Apr 17, 2017 8:33 pm

Mallika Prova wrote:surely it wants to be $MN||AO$ ??


Thanks :oops: I edited that.
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