2010 China Second Round,test 2,problem 1

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2010 China Second Round,test 2,problem 1

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#1 h Feb 11, 2012, 5:24 AM • 2 Y

Y by Adventure10, Mango247

Given an acute triangle whose circumcenter is $O$ .let $K$ be a point on $BC$ ,different from its midpoint. $D$ is on the extension of segment $AK,BD$ and $AC$ , $CD$ and $AB$ intersect at $N,M$ respectively.prove that $A,B,D,C$ are concyclic.

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#2 h Feb 11, 2012, 8:22 AM • 2 Y

Y by Adventure10, Mango247

littletush wrote:

Given an acute triangle whose circumcenter is $O$ .let $K$ be a point on $BC$ ,different from its midpoint. $D$ is on the extension of segment $AK,BD$ and $AC$ , $CD$ and $AB$ intersect at $N,M$ respectively.prove that $A,B,D,C$ are concyclic.

are you sure you've posted the problem correctly?? I can't see what's the usage of $M,N,O$ , and $D$ can be chosen independently....

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#3 h Feb 11, 2012, 9:55 AM • 1 Y

Y by Adventure10

Constructing all points given, we don't have any circle through $4$ points.
( looking to usage, we would have there is an other point on $\odot MON$ seems wrong with my picture)

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#4 h Feb 11, 2012, 9:58 AM • 1 Y

Y by Adventure10

I presume missing $OK\bot MN$ .

Best regards,
sunken rock

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#5 h Feb 11, 2012, 10:49 AM • 2 Y

Y by Adventure10, Mango247

Indeed, with that property, if $ABCD$ is concyclic,
$MN$ is the polar of $K$ wrt $(O).$
( known lemma, but I never asked the proof)
Second thing, for fixed $K$ and an other $D$ we see $M'N'$ isn't parallel to $MN.$
( proof for this with Ceva)

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#6 h Jun 1, 2016, 4:13 AM • 1 Y

Y by Adventure10

sunken rock wrote:

I presume missing $OK\bot MN$ .

Best regards,
sunken rock

yes,he missed it

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#7 h Oct 26, 2022, 3:38 AM

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The problem would be much easier if we prove $OK \bot MN$ with $\odot(ABCD)$ given. But I can't figure out how to adjust the definition order.

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#8 h Apr 28, 2024, 10:21 PM

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As shown in the figure, the circumcenter of acute triangle $ABC$ is $O$ , $K$ is a point on side $BC$ (not the midpoint of side $BC$ ), $D$ is a point on the extension of line segment $AK$ , straight line $BD$ intersects $AC$ at point $N$ , and straight line $CD$ intersects $AB$ at point $M$ . Prove that if $OK\perp MN$ , then the points $A$ , $B$ , $D$ and $C$ lie on a circle.

https://cdn.artofproblemsolving.com/attachments/8/b/1478070c02f2a8b2191efc57e7dfb754929681.png

This post has been edited 3 times. Last edited by parmenides51, Apr 29, 2024, 5:13 PM

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