Iran 2008 Geometry

by utkarshgupta, Jan 31, 2015, 12:07 PM

An easy problem as a refresher before INMO

Problem :
Let $I_a$ be the $A$ -excenter of $\Delta ABC$ and the $A$ -excircle of $\Delta ABC$ be tangent to the lines $AB,AC$ at $B',C'$ , respectively. $I_aB,I_aC$ meet $B'C'$ at $P,Q$ , respectively. $M$ is the meet point of $BQ,CP$ . Prove that the length of the perpendicular from $M$ to $BC$ is equal to $r$ where $r$ is the radius of incircle of $\Delta ABC$ .

Solution :
Let the given excircle be tangent to $BC$ at $T$ .
Let $\angle ABC = \angle B$ , $\angle ACB =\angle C$ and $\angle BAC =\angle A$

Then it is easy to see by simple angle chasing that
$PI_ACC'T$ , $QI_ABB'T$ and $BCQP$ are cyclic.

Radical Axis of $\odot PI_ACC'T$ and $\odot QI_ABB'T$ is $TI_A$
Since $BCQP$ are cyclic, $TMI_A$ is a straight line.

$\angle MTC = \angle I_ATC =90$

Also since $M$ lies on the radical axis,
$MT \cdot MI_A = MC \cdot MP$
$\implies CTPI_A$ is cyclic
$\implies \angle CPI_A = 90$
$\implies \angle TCP = \frac{\angle B}{2}$
$\implies MT = CT \tan {\frac{B}{2}}$
$\implies MT = (s-b)\tan {\frac{B}{2}} = r$

QED

geometry

trigonometry

Iran

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here is my solution: I will written the problem: ABC is a..triangle (I) , (O) ,blah blah .we have BB'C=BIaC => BQ pedendicular CIa and analogously CP pedendicular BIa => X is othorcentre BCIa .associate IBIaC concylic with IIa is a diameter then BICX paralegrogram ..

by lebathanh, Aug 25, 2016, 3:13 AM

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Here goes first post of 2025! Great blog.
by math_holmes15, Jan 14, 2025, 8:53 AM
First post of 2024
by Yiyj1, Feb 8, 2024, 5:40 AM
First post of 2023
by HoRI_DA_GRe8, Jul 22, 2023, 7:45 AM
Nice blog ! Your isogonality lemma is really powerful !
by 554183, Oct 14, 2021, 8:55 AM
Post plss....
by samrocksnature, Apr 11, 2021, 10:12 PM
alas,this is ded
by Hamroldt, Mar 18, 2021, 4:13 PM
Thanks for the nice blog.
by Feridimo, Mar 6, 2020, 4:17 PM
I think this might be silly but ... when should we expect to have another post ?? I am very keen to see it
by gamerrk1004, Nov 4, 2019, 4:54 PM
Let's all echo what's written in the blog description - Stay Insane / 'Cause it's your labor, will and pain/ That takes you to the top of soda fountain
by Kayak, Oct 2, 2017, 7:18 PM
hey utkarsh jee is over now ... continue your elementary blog pleaseeeeeee!
by kk108, Jun 17, 2017, 11:19 AM
Congrats on becoming a contest moderator!
by Ankoganit, Mar 9, 2017, 5:22 AM
INTERSTING BLOG
by kk108, Feb 19, 2017, 2:04 PM
I have no plans for this blog right now....
No time here people !
But lets see....
I may try some combinatorics
by utkarshgupta, Feb 15, 2017, 12:47 PM
Thanks for the nice blog!
by Orkhan-Ashraf_2002, Feb 13, 2017, 6:34 PM
Revive it!!!
Best blog out there, for sure!
by rmtf1111, Jan 12, 2017, 6:02 PM
Oh you certainly will..
Bu I don't think I will
by achilles04, Feb 11, 2015, 6:24 PM
Thanks
Hope you are there too ...

And hope I get selected
by utkarshgupta, Feb 11, 2015, 11:14 AM
Nice blog ..
keep it up and you might one day become a mathematician lIke me
( just joking )
Btw don't forget us after going to imotc.
by achilles04, Feb 10, 2015, 4:49 PM
nice blog!!!!!
by pradhit, Feb 7, 2015, 11:51 AM
No ideas about the cut off but should be less than 60 according to me...
So you never know
by utkarshgupta, Feb 6, 2015, 4:06 PM
Hi utkarsh,
what do u expect the cutoff to be this year??

by kgdpsdurg, Feb 6, 2015, 12:41 PM
Hi utkarsh
How many did you clear in INMO 2015?
by moldevort, Feb 1, 2015, 3:59 PM
@ Anonymous Bunny
If we both (that should have included me) are lucky enough !
by utkarshgupta, Sep 17, 2014, 2:44 AM
Nice blog. Great job!

If I am lucky enough, hopefully we shall meet at IMOTC.
by AnonymousBunny, Sep 16, 2014, 8:08 PM
Thanx all !!

If anyone wish to contribute anything just pm !

I will add u to the contributors' list !
by utkarshgupta, Sep 16, 2014, 6:35 AM
doing good progress!! btw nice blog.
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Nice blog.
by Kezer, Aug 5, 2014, 4:04 PM
Hi Utkarsh! Wonderful blog. Keep it up.
by YESMAths, Jul 20, 2014, 1:36 PM
I need some shouts and characters

by utkarshgupta, Jul 20, 2014, 4:22 AM

48 shouts

Elementary

Iran 2008 Geometry

by utkarshgupta, Jan 31, 2015, 12:07 PM

1 Comment