[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 - IMO Shortlist 2010 G7

## IMO Shortlist 2010 G7

For discussing Olympiad level Geometry Problems

### IMO Shortlist 2010 G7

Three circular arcs $\gamma_1, \gamma_2,$ and $\gamma_3$ connect the points $A$ and $C.$ These arcs lie in the same half-plane defined by line $AC$ in such a way that arc $\gamma_2$ lies between the arcs $\gamma_1$ and $\gamma_3.$ Point $B$ lies on the segment $AC.$ Let $h_1, h_2$, and $h_3$ be three rays starting at $B,$ lying in the same half-plane, $h_2$ being between $h_1$ and $h_3.$ For $i, j = 1, 2, 3,$ denote by $V_{ij}$ the point of intersection of $h_i$ and $\gamma_j$ (see the Figure below). Denote by $\widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}}$ the curved quadrilateral, whose sides are the segments $V_{ij}V_{il},$ $V_{kj}V_{kl}$ and arcs $V_{ij}V_{kj}$ and $V_{il}V_{kl}.$ We say that this quadrilateral is $circumscribed$ if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals $\widehat{V_{11}V_{21}}\widehat{V_{22}V_{12}}, \widehat{V_{12}V_{22}}\widehat{V_{23}V_{13}},\widehat{V_{21}V_{31}}\widehat{V_{32}V_{22}}$ are circumscribed, then the curved quadrilateral $\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}$ is circumscribed, too.
Attachments
2010 G7.png (24.2 KiB) Viewed 343 times
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.

Thanic Nur Samin

Posts: 176
Joined: Sun Dec 01, 2013 11:02 am

### Re: IMO Shortlist 2010 G7

Lemma : Let two circular arcs $\alpha$ & $\beta$ connect pionts $A,B$. If two circle $\varpi_1$ & $\varpi_2$ are tangent to $\alpha$ & $\beta$ ,then their external center of simillitude lies in $AB$.

Proof :Let an external common tangent $l$ of $\alpha$ & $\beta$ meet $AB$ at piont $M$.Then the invertion with center $M$ ,taking $\alpha$ to itself ,takes $\beta$ to itself.Thus it takes $\varpi_1$ to a circle tangent to $l$ , $\alpha$ & $\beta$ ,but there can be atmost two circles tangent to two circular arcs and a line and lying in the same side of the line . So the invertion takes $\varpi_1$ to $\varpi_2$,so $M$ is their external center of simillitude.

Solution :Let $O_1,O_2,O_3$ be the incenters of $\widehat{V_{11}V_{21}}\widehat{V_{22}V_{12}}, \widehat{V_{12}V_{22}}\widehat{V_{23}V_{13}},\widehat{V_{21}V_{31}}\widehat{V_{32}V_{22}}$ , respectively. Let $(O_4)$ be the circle tangent to $h_2 ,\gamma _2,\gamma _3$ such that $(O_3),(O_4)$ lies in the sane side of $h_2$. Let $O_1O_3 \cap h_2=X.O_2O_4 \cap h_2=Y$.$O_1O_3 \cap AC=Z$. Then $X$ & $Z$ are the internal and external center of simillitude of $(O_1),(O_3)$.So $(BA,h_2 ;BO_1,BO_3)=-1$. Similerly $(BA,h_2 ;BO_2,BO_4)=-1$ .As $B,O_1,O_2$ are collinear so $B,O_3,O_4$ are collinear implying that $(O_4)$ is tangent to $h_3$ thus it is the incircle of $\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}$.
The more I learn, the more I realize how much I don't know.

- Albert Einstein
joydip

Posts: 42
Joined: Tue May 17, 2016 11:52 am