[MOM] What I do home alone on a Sunday night...

by shiningsunnyday, Sep 11, 2016, 4:39 PM

Mongolian TST / Lemmas in Oly Geo wrote:

Let $O$ be the circumcenter of the acute-angled triangle $ABC,$ and let $M$ be a point on the circumcircle of triangle $ABC.$ Let $X, Y, Z$ be the projections of $M$ onto $OA, OB, OC,$ respectively. Prove that the incenter of triangle $XYZ$ lies on the Simson line of $M$ with respect to triangle $ABC.$

Solution
Tidbit

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This post has been edited 1 time. Last edited by shiningsunnyday, Sep 11, 2016, 4:40 PM

geometry

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You are good at drawing and your handwriting is neat... things I have never managed to nail down

by zephyrcrush78, Sep 11, 2016, 5:39 PM

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ayy I think I see it

homothety at M, then show that I under homothety still lies on the line.

Yea that's a nice first step. But we don't know much about the image of I, so what you gonna do next birdie (though I kind of drew it in the diagram oops).

This post has been edited 1 time. Last edited by shiningsunnyday, Sep 12, 2016, 5:55 PM

by agbdmrbirdyface, Sep 11, 2016, 6:28 PM

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ssd

please sleep

i’m concerned

by cjquines0, Sep 12, 2016, 1:38 AM

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hm i think i have a sol without using fact that simson line bisects $MH$

note

Hm that's an interesting observation; I'll see where I can get with it sometime in the near future; also my words in the margin lead to a dead end lol; I ended up using an entirely different approach

This post has been edited 2 times. Last edited by shiningsunnyday, Sep 12, 2016, 5:58 PM

by tastymath75025, Sep 12, 2016, 3:38 AM

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this guy is an absolute legend. much love wherever you are Michael
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amazing blog
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hi i randomly found this
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can i be a contributor please?
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by samrocksnature, Jul 16, 2021, 8:25 PM
2021 post
by the_mathmagician, May 5, 2021, 3:28 PM
Let $ABC$ be an equilateral triangle of side length $1$ . Let $D$ be the point such that $C$ is the midpoint of $BD$ , and let $I$ be the incenter of triangle $ACD$ . Let $E$ be the point on line $AB$ such that $DE$ and $BI$ are perpendicular. $ \
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Nice blog!
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contrib /charmander
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all contrib requests added I usually get to posting every night after running on the treadmill and showering, posting my favorite problem(s) I did during the day.
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Since problems is plural, he might post multiple problems through different posts in one day...
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hi can i have contrib?
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Hello it is called problems of the DAY
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How often will this blog be updated?
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why are we counting shouts

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Though probably not because it's not like I'll post anything anyways :/ :/
by zephyrcrush78, Jun 15, 2016, 1:20 AM
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new goal: get the shouts that are congruent to 1 mod 5
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Problems of the day

[MOM] What I do home alone on a Sunday night...

by shiningsunnyday, Sep 11, 2016, 4:39 PM

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