Right Angles on Incircle Chord

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Thamim Zahin
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Right Angles on Incircle Chord

Unread post by Thamim Zahin » Thu Aug 04, 2016 11:47 pm

The incircle of $\triangle ABC$ with incentre $I$ touches the sides $BC,CA$ and $AB$ at $D,E$ and $F$ respectively. Now, let $K=BI\cap EF$. Show that $BK\perp CK$.
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Thanic Nur Samin
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Re: Right Angles on Incircle Chord

Unread post by Thanic Nur Samin » Fri Aug 05, 2016 12:04 am

Hint:
$IEKCD$ is cyclic.
Bonus problem: Let the reflection of $C$ through $K$ be $C_1$. Prove that $A,B$ and $C_1$ are collinear.
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tanmoy
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Re: Right Angles on Incircle Chord

Unread post by tanmoy » Fri Aug 05, 2016 12:15 am

Let $EF \cap BC=L$.Then $(C,B;D,L)=-1$ and $\angle LKB=\angle BKD$,So,$BK \perp KC$.
Thanic Nur Samin wrote: Bonus problem: Let the reflection of $C$ through $K$ be $C_1$. Prove that $A,B$ and $C_1$ are collinear.
Let $CK \cap AB=C_{1}$.$\Delta C_{1}BK$ is the reflection of $\Delta CBK$ across $BK$.So,the reflection of $C$ through $K$ is $C_1$.
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