AoPS - Problem

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Problem

Source: Turkey TST 2022 P4 Day 2

Tags: geometry, geometry proposed, circles, tangent circles

electrovector

13.03.2022 14:38

We have three circles $w_1$ , $w_2$ and $\Gamma$ at the same side of line $l$ such that $w_1$ and $w_2$ are tangent to $l$ at $K$ and $L$ and to $\Gamma$ at $M$ and $N$ , respectively. We know that $w_1$ and $w_2$ do not intersect and they are not in the same size. A circle passing through $K$ and $L$ intersect $\Gamma$ at $A$ and $B$ . Let $R$ and $S$ be the reflections of $M$ and $N$ with respect to $l$ . Prove that $A, B, R, S$ are concyclic.

electrovector

13.03.2022 14:39

Let

$O_1, O, O_2$ be the centers of

$w_1, w_2$ and

$\Gamma$ .

$\angle MKL= \frac{\angle KO_1M}{2}=\frac{360-\angle MON-\angle NO_2L}{2}=\angle ONM+\angle O_2NL$ which proves that

$KMNL$ is concyclic. Also we have that

$ABLK$ and

$MNBA$ are concyclic which means

$AB$ ,

$KL$ and

$MN$ intersects at some point,

$T$ . It is also easy to see that

$MN$ ,

$RS$ and

$KL$ intersect at some point which means

$T \in RS$ . Now we have

$TR \cdot TS= TM \cdot TN= TA \cdot TB$ which means

$ABSR$ is concyclic.

Attachments:

badumtsss

20.03.2022 17:28

Because of my amazing geometry skills, I couldn't see the direct angle chasing. Anyway, by Monge Theorem, we know that the center of homotety sending $\omega_1$ to $\omega_2$ is the intersection of $KL$ and $MN$ . Take the second intersections $X$ , $Y$ of $MN$ and $\omega_1$ , $\omega_2$ , angle chasing shows that $KLXY$ is cyclic, which also shows that $KLMN$ is cyclic. $MN$ and $RS$ intersect on $KL$ which ends the proof.