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New geo: prove $OM$ passes through midpoint
Let $ABC$ be an equilateral triangle of center $O$, and $M\in BC$. Let $K,L$ be projections of $M$ onto the sides $AB$ and $AC$ respectively. Prove that line $OM$ passes through the midpoint of the segment $KL$.
Source:
Source:
Last edited by Moon on Sun Dec 19, 2010 10:34 pm, edited 1 time in total.
Reason: Corrected!
Reason: Corrected!
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
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Re: New geo: prove $OM$ passes through midpoint
Did you mean segment $KL$? :S
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter
Re: New geo: prove $OM$ passes through midpoint
Yup...you are right. Corrected!
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
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Re: New geo: prove $OM$ passes through midpoint
Join $AM$.
Let $P$ be the midpoint of $AM$ and $R$ be the feet of perpendicular from $A$ to $BC$.
Now $A, K, M, R, L$ are concyclic with $AM$ as diameter. Join $P$ and $R$. Now $PK=PL=PR=PM=AP$= the radius of the circle. $\angle KPR$ = $2\angle KAR$ = 2X30 and as $PK= PR$, $\angle PKR = \angle PRK$ = $120/2 = 60$.
PKRL is a rhombus. $PR$ and $KL$ being the diagonals of a rhombus are perpendicular to each other and $Q$ is the midpoint of $KL$. Now, join $MQ$ and extend it to meet $AR$ at $T$.
As $PQ=QR$, $\triangle PMQ=\triangle QMR$ and also $\triangle PTQ= \triangle TQR$. So, $\triangle PTM$ = $\triangle TMR$
Again as $AP=PM$, $\triangle MPT$ = $\triangle APT$, $\triangle ATM$ = $\triangle PTM$
So, \[ \frac {\triangle TRM}{\triangle ATM} = \frac {1}{2} \] but we know, $\frac {OR}{AT} = \frac {1}{2}$
This implies that $T$ and $O$ is the same point. $OM$ intersects $KL$ at $Q$ which is the midpoint of $KL$.
Let $P$ be the midpoint of $AM$ and $R$ be the feet of perpendicular from $A$ to $BC$.
Now $A, K, M, R, L$ are concyclic with $AM$ as diameter. Join $P$ and $R$. Now $PK=PL=PR=PM=AP$= the radius of the circle. $\angle KPR$ = $2\angle KAR$ = 2X30 and as $PK= PR$, $\angle PKR = \angle PRK$ = $120/2 = 60$.
PKRL is a rhombus. $PR$ and $KL$ being the diagonals of a rhombus are perpendicular to each other and $Q$ is the midpoint of $KL$. Now, join $MQ$ and extend it to meet $AR$ at $T$.
As $PQ=QR$, $\triangle PMQ=\triangle QMR$ and also $\triangle PTQ= \triangle TQR$. So, $\triangle PTM$ = $\triangle TMR$
Again as $AP=PM$, $\triangle MPT$ = $\triangle APT$, $\triangle ATM$ = $\triangle PTM$
So, \[ \frac {\triangle TRM}{\triangle ATM} = \frac {1}{2} \] but we know, $\frac {OR}{AT} = \frac {1}{2}$
This implies that $T$ and $O$ is the same point. $OM$ intersects $KL$ at $Q$ which is the midpoint of $KL$.
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter
Re: New geo: prove $OM$ passes through midpoint
ইমেজ অ্যাটাচ করতে পারতেসি না।
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter
Re: New geo: prove $OM$ passes through midpoint
ইমেজ এটাচ না করতে পারার কারণ কী? কী লেখা আসে?
আমি তো কোন সমস্যা দেখতিছিনা।
আমি তো কোন সমস্যা দেখতিছিনা।
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: New geo: prove $OM$ passes through midpoint
কি জানি। :S
I will check again
I will check again
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter