Looking for a synthetic solution

For discussing Olympiad level Geometry Problems
User avatar
Thanic Nur Samin
Posts:176
Joined:Sun Dec 01, 2013 11:02 am
Looking for a synthetic solution

Unread post by Thanic Nur Samin » Thu Aug 04, 2016 5:13 pm

$\triangle ABC$ is a triangle, such that $AB\neq AC$. The incircle of $\triangle ABC$ touches $BC,CA,AB$ at
$D,E,F$ respectively. $H$ is a point on the segment $EF$ such that $DH\perp EF$. Suppose $AH\perp BC$, prove that $H$ is the orthocentre of $\triangle ABC$.
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.

tanmoy
Posts:312
Joined:Fri Oct 18, 2013 11:56 pm
Location:Rangpur,Bangladesh

Re: Looking for a synthetic solution

Unread post by tanmoy » Thu Aug 04, 2016 11:59 pm

A synthetic solution:
Lemma: $\triangle ABC$ is a triangle, such that the incircle of $\triangle ABC$ touches $BC, CA, AB$ at $D, E, F$ respectively. $H$ is a point on the segment $EF$ such that $DH\perp EF$. And let $P$ be the antipode of $A$. Then $H, I, P$ are collinear.
Proof: Let $O$ be the center of the circumcircle of $\triangle ABC$. Let $AI \cap \odot(O)$=$Q$ and $AI \cap EF=R$. As $PQ$ is parallel to $HR$, if we can show that $\displaystyle{\frac{HR}{QP}=\frac{RI}{IQ}}$, we are done. To show this, let $QO \cap \odot(O)=S$.Then $REI \sim PSQ$. Let $T$ is the projection of $D$ on $AI$. Then $IDT \sim AQP$. So, \[\frac{RI}{IQ}=\frac{RI}{QC}=\frac{IE}{QS}=\frac{IE}{AP}=\frac{ID}{AP}=\frac{DT}{QP}=\frac{HR}{QP}\]Therefore, $H, I, P$ are collinear. The lemma is proved.

Assume, $X$ is the antipode of $D$ in $(I)$ and let $M$ be the midpoint of $BC$. Then it is well known that $IM$ is parallel to $AX$..........(1)
$AHDI$ is a parallelogram. So, $AH=DI=IX$ and $AH$ is parallel to $IX$. So, $AHIX$ is also a parallelogram. So, $HI$ is parallel to $AX$. Then from (1), $H,I,M$ are collinear.
And from the lemma, $H,I,P$ are collinear. So, $H,M,P$ are collinear. Then it is well known that $H$ is the orthocenter of $\triangle ABC$. :mrgreen:
Last edited by Phlembac Adib Hasan on Fri Aug 05, 2016 8:17 am, edited 1 time in total.
Reason: Fixed spacing and made minor changes to the latex code
"Questions we can't answer are far better than answers we can't question"

Post Reply