Looooong Geo Finale for Day 2

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Looooong Geo Finale for Day 2

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Source: Turkey TST 2025 P6

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#1 h Mar 18, 2025, 6:44 AM

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Let $ABC$ be a scalene triangle with incenter $I$ and incircle $\omega$ . Let the tangency points of $\omega$ to $BC,AC\text{ and } AB$ be $D,E,F$ respectively. Let the line $EF$ intersect the circumcircle of $ABC$ at the points $G, H$ . Assume that $E$ lies between the points $F$ and $G$ . Let $\Gamma$ be a circle that passes through $G$ and $H$ and that is tangent to $\omega$ at the point $M$ which lies on different semi-planes with $D$ with respect to the line $EF$ . Let $\Gamma$ intersect $BC$ at points $K$ and $L$ and let the second intersection point of the circumcircle of $ABC$ and the circumcircle of $AKL$ be $N$ . Prove that the intersection point of $NM$ and $AI$ lies on the circumcircle of $ABC$ if and only if the intersection point of $HB$ and $GC$ lies on $\Gamma$ .

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#2 h Wednesday at 10:33 PM • 3 Y

Y by ehuseyinyigit, egxa, AnSoLiN

First we establish some properties of the configuration:

Property I: If $R$ is the foot of $D$ onto $EF$ , then $M$ lies on segment $AR$
proof
Property II: If $P$ is the intersection of $EF$ and $BC$ then $P$ lies on $AN$ and $D$ lies on the polar of $P$ with respect to $(ABC)$
proof
Property III: Let $AI$ meet $(ABC)$ again at $M_a$ , then the intersection of $PM_a$ and $AM$ lies on $(ABC)$
proof

To solve the problem we show that both statements are equivalent to $M$ lying on the polar of $P$ with respect to $(ABC)$ :

Part I: $N$ , $M$ , and $M_a$ are collinear if and only if $M$ lies on the polar of $P$ with respect to $(ABC)$
proof
Part II: The intersection point of $HB$ and $GC$ lies on $\Gamma$ if and only if $M$ lies on the polar of $P$ with respect to $(ABC)$
proof

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This post has been edited 4 times. Last edited by sami1618, Wednesday at 10:38 PM

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