[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include(/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php) [function.include]: failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include() [function.include]: Failed opening '/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php' for inclusion (include_path='.:/opt/php53/lib/php')
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include(/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php) [function.include]: failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include() [function.include]: Failed opening '/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php' for inclusion (include_path='.:/opt/php53/lib/php')
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include(/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php) [function.include]: failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include() [function.include]: Failed opening '/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php' for inclusion (include_path='.:/opt/php53/lib/php')
[phpBB Debug] PHP Warning: in file [ROOT]/includes/session.php on line 1042: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4786: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4788: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4789: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4790: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
BdMO Online Forum • View topic - IGO 2016 Advanced/3

IGO 2016 Advanced/3

For discussing Olympiad level Geometry Problems
Facebook Twitter

IGO 2016 Advanced/3

Post Number:#1  Unread postby Thamim Zahin » Tue Jan 10, 2017 4:24 pm

3. Let $P$ be the intersection point of sides $AD$ and $BC$ of a convex qualrilateral $ABCD$. Suppose that $I_1$ and $I_2$ are the incenters of triangles $PAB$ and $PDC$, respectively. Let $O$ be the circumcenter of $PAB$, and $H$ the orthocenter of $PDC$. Show that the circumcircles of triangles $AI_1B$ and $DHC$ are tangent together if and only if the circumcircles of triangles $AOB$ and $DI_2C$ are tangent together.
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.
User avatar
Thamim Zahin
 
Posts: 98
Joined: Wed Aug 03, 2016 5:42 pm

Re: IGO 2016 Advanced/3

Post Number:#2  Unread postby joydip » Fri Jan 13, 2017 1:10 pm

A generalization of the problem :

Let $P$ be the intersection point of sides $AD$ and $BC$ of a convex qualrilateral $ABCD$. Suppose that $I_1$ and $I_2$ are the incenters of $\triangle PAB $ & $\triangle PDC$, respectively. Let $O$ be the circumcenter of $\triangle PAB$, and $H $ the orthocenter of $\triangle PDC$. Show that the circumcircles of $\triangle AI_1B$ and $\triangle DHC$ intersects at an angle $\theta$ if and only if the circumcircles of $\triangle AOB $ and $\triangle DI_2C $ intersects at an angle $\theta$.

Proof:
Lemma :
Let $\alpha$ & $ \beta$ be two circles meeting at point $A , B$. If $C \in \alpha$ & $ D \in \beta$, then the angle between $\alpha$ & $ \beta$ is equal to the angle between $(ADC)$ & $(BDC)$.

proof :Let an inversion with center $D$ and an arbitary radius take $A,B,C$ to $A',B',C'$ respectively .Then the angle between $A'C'$ & $B'C'$ is equal to the angle between $A'B'$ & $(A'B'C')$ by alternate segment theorem ,completing the proof .

Let $(AI_1B),(DI_2C),(DHC)$ meet $BC$ at $M,N,S$ respectively .Let $(AI_2B) \cap (DHC)={ U,V }$.
Let $(BVC) $ & $(AVD)$ meet again at $X$.
$\angle DXC= \angle VAD +\angle VBC=\angle VAD + \angle MAV=\angle MAD=\angle DNC $ So, $X \in (DI_2C)$

$\angle AXB= \angle VXB + \angle AXV= \angle VCB + \angle VDA=\angle SDV + \angle VDA= \angle SDA=2\angle APB=\angle AOB $. So, $X \in (AOB)$
Let $(BUC) $ & $(AUD)$ meet again at $Y$.Then Similerly we get $(DI_2C) \cap (AOB) ={X,Y}$
So, by our lemma angle between $(AUV)$ & $(DUV)$ , angle between $(AUD)$ & $(AVD)$,and angle between $(AXY)$ & $(DXY)$ are same, completing the main proof.

(the original problem can be solved more easily, only by considering $X $ & $Y$)
The more I learn, the more I realize how much I don't know.

- Albert Einstein
joydip
 
Posts: 39
Joined: Tue May 17, 2016 11:52 am


Share with your friends: Facebook Twitter

  • Similar topics
    Replies
    Views
    Author

Return to Geometry

Who is online

Users browsing this forum: Baidu [Spider] and 2 guests