Circle is tangent to circumcircle and incircle

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Kazi_Zareer
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Circle is tangent to circumcircle and incircle

Unread post by Kazi_Zareer » Sat Feb 04, 2017 12:19 pm

In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$. Finally, let $\gamma$ be the circumcircle of $\triangle AST$.

(a) Show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$.

(b) Show that $\gamma$ is tangent to the incircle of $\triangle ABC$.
We cannot solve our problems with the same thinking we used when we create them.

rah4927
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Re: Circle is tangent to circumcircle and incircle

Unread post by rah4927 » Sat Feb 04, 2017 6:48 pm

Without loss of generality, let $Y$ be closer to $A$ than $X$. The crux move is to show that $AD$ bisects $\angle SAT$. Let $L$ be the midpoint of $XY$. Clearly $X,Y,S,T$ all lie on a circle centered at $L$. Denote this circle by $\ell$.

Lemma : $CY\perp AX$ and $BX\perp AX$

Proof : Well known. Chase angles.

Lemma : The incircle and $\ell$ are orthogonal.

Proof : It suffices to show that $IY\cdot IX=r^2=IE^2$. Let $IC$ intersect $ED$ at $M$. Then since $\angle XYC = \angle XMC =90^{\circ}$, $XYMC$ is a cyclic quadrilateral. Therefore $IY\cdot IX = IM\cdot IC = IE^2$. We are done.

Lemma : $LA$ bisects $\angle SAT$.

Proof : Being radius of the same circle, $LS=LT$. Therefore it suffices to prove that $L$ lies on the circumcircle of $\triangle AST$.Let $AX$ intersect $BC$ at $P$. It therefore suffices to show that $LP\cdot LA=LT^2=LY^2$.

$$XY=AX-AY=c\cos\dfrac{A}{2} -b\cos\dfrac{A}{2}$$

implying $LY=\dfrac{1}{2}XY = \dfrac12(c-b)\cos\dfrac{A}{2}$.

$$LA=LY+YA=\dfrac12(b+c)\cos\dfrac{A}{2}$$

meaning $LP\cdot LA = (LA-AP)\cdot LA = LA^2 - LA\cdot AP = \dfrac14(b+c)^2\cos\dfrac{A}{2}-\dfrac12(b+c)\cos\dfrac{A}{2}\cdot \dfrac{2bc}{b+c}\cos\dfrac{A}{2}$ which calculates to $\dfrac14(\cos\dfrac{A}{2})^2(b-c)^2=LY^2$
thus ending the proof.

For part (a), let the circumcircle of $AST$ intersect $AC$ and $AC$ at $W$ and $Z$, It suffices to show that $WZ$ is parallel to $BC$, which is obvious since by our previous lemma, $\angle WAS = \angle TAZ$.

For part (b), invert with respect to $\ell$. This leaves the incircle right where it is, but takes the circumcircle of $AST$ to line $BC$, since $L$ also lies on circle $AST$. Since $BC$ and the incircle are tangent, we are done.
Last edited by rah4927 on Sun Feb 05, 2017 12:35 am, edited 3 times in total.

Absur Khan Siam
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Re: Circle is tangent to circumcircle and incircle

Unread post by Absur Khan Siam » Sat Feb 04, 2017 6:57 pm

Will bisector $\angle BAC$ intersect both $DE$ and $DF$?
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Kazi_Zareer
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Re: Circle is tangent to circumcircle and incircle

Unread post by Kazi_Zareer » Sat Feb 04, 2017 8:58 pm

Absur Khan Siam wrote:Will bisector $\angle BAC$ intersect both $DE$ and $DF$?
Extend $DF$ and it will intersect the bisector of $\angle BAC$ at $Y$.
We cannot solve our problems with the same thinking we used when we create them.

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