Right Angles on Incircle Chord
- Thamim Zahin
- Posts:98
- Joined:Wed Aug 03, 2016 5:42 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
- Posts:176
- Joined:Sun Dec 01, 2013 11:02 am
Re: Right Angles on Incircle Chord
Hint:
Bonus problem: Let the reflection of $C$ through $K$ be $C_1$. Prove that $A,B$ and $C_1$ are collinear.
Hammer with tact.
Because destroying everything mindlessly isn't cool enough.
Because destroying everything mindlessly isn't cool enough.
Re: Right Angles on Incircle Chord
Let $EF \cap BC=L$.Then $(C,B;D,L)=-1$ and $\angle LKB=\angle BKD$,So,$BK \perp KC$.
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 $C_1$.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.
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