ISL 2002 G3

by Wolstenholme, Aug 4, 2014, 2:50 AM

The circle $S$ has centre $O$ , and $BC$ is a diameter of $S$ . Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$ . Let $D$ be the midpoint of the arc $AB$ which does not contain $C$ . The line through $O$ parallel to $DA$ meets the line $AC$ at $I$ . The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$ . Prove that $I$ is the incentre of the triangle $CEF.$

Proof:

Since $AE = AF$ we have that arcs $\widehat{AE} = \widehat{AF}$ so $CI$ bisects $\angle{ECF}$ . Therefore, since $A$ is the midpoint of arc $\widehat{EF}$ of circle $S$ , it suffices to show that $AE = AF = AI$ . Now since segments $EF$ and $AO$ are perpendicular and bisect one another we have that quadrilateral $AEOF$ is a rhombus so $AE = AF = OF = OD$ so it suffices to show that $AI = OD$ so it suffices to show that $AIOD$ is a parallelogram. But $DO \perp AB$ by definition and $AI \perp AB$ since $BC$ is a diameter of $S$ so $AIOD$ is in fact a parallelogram and we are done.

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glad to have found a fellow chipotle lover <3
by nukelauncher, Aug 13, 2020, 6:40 AM
the random chinese tst problem is the only thing I read, but I'll assume your blog is nice and give you a shout even though you probably never use aops anymoer
by fukano_2, Jun 14, 2020, 6:24 AM
wolstenholme - op
by AopsUser101, Jan 29, 2020, 8:27 PM
this blog is so hot
by mathleticguyyy, Jun 5, 2019, 8:26 PM
Hi. Nice Blog!
by User360702, Jan 10, 2019, 6:03 PM
helloooooo
by songssari, Jun 12, 2016, 8:21 AM
shouts make blogs happier
by briantix, Mar 18, 2016, 9:57 PM
You were just featured on AoPS's facebook page.
by mishka1980, Sep 12, 2015, 10:33 PM
This is late, but where is the ARML results post?
by donot, Aug 31, 2015, 11:07 PM
"I am Sam"
"Sam I am"
by mathwizard888, Aug 12, 2015, 9:13 PM
HW $\textcolor{white}{}$
by Eugenis, Apr 20, 2015, 10:10 PM
Uh-oh ARML practice is Thursday... I should start the homework.
by nosaj, Apr 20, 2015, 12:34 AM
Yes I am Sam, and Chebyshev polynomials aren't trivial, although they do make some problems trivial
by Wolstenholme, Apr 15, 2015, 10:00 PM
How are Chebyshev Polynomials trivial?
by nosaj, Apr 13, 2015, 4:10 AM
Are you Sam?
by Eugenis, Apr 4, 2015, 2:05 AM
@Brian: yes, yes I did #whoneedsalgskillz?

@gauss1181; hey!
by Wolstenholme, Mar 1, 2015, 11:25 PM
hello!!!
by gauss1181, Nov 27, 2014, 12:19 AM
Hi Wolstenholme did you actually use calc on that tstst problem
by briantix, Aug 2, 2014, 12:25 AM

18 shouts

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ISL 2002 G3

by Wolstenholme, Aug 4, 2014, 2:50 AM

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