Geo with two circles and a variable line

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Geo with two circles and a variable line

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Source: Czech-Polish-Slovak Match 2022 P3

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#1 h Sep 3, 2022, 3:03 PM • 3 Y

Y by Han-Ji-Seung, TemetNosce, Mango247

Circles $\Omega_1$ and $\Omega_2$ with different radii intersect at two points, denote one of them by $P$ . A variable line $l$ passing through $P$ intersects the arc of $\Omega_1$ which is outside of $\Omega_2$ at $X_1$ , and the arc of $\Omega_2$ which is outside of $\Omega_1$ at $X_2$ . Let $R$ be the point on segment $X_1X_2$ such that $X_1P = RX_2$ . The tangent to $\Omega_1$ through $X_1$ meets the tangent to $\Omega_2$ through $X_2$ at $T$ . Prove that line $RT$ /is tangent to a fixed circle, independent of the choice of $l$ .

This post has been edited 1 time. Last edited by a_507_bc, Sep 3, 2022, 3:04 PM

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#2 h Sep 3, 2022, 4:02 PM • 1 Y

Y by TemetNosce

Let the intersection point of $\Omega _1$ and $\Omega _2$ be $Q \neq P$ .
Since $\angle X_1 T X_2 = 180 - \angle T X_1 X_2 - \angle T X_2 X_1 = 180 - \angle X_1 Q P - \angle X_2 Q P = 180 - \angle X_1 Q X_2$ ,
$\angle X_1 Q X_2 + \angle X_1 T X_2 = 180$ , therefore $(X_1, T, X_2, Q)$ are cyclic.
Let the circle that passes through $X_1, T, X_2, Q$ be $\omega$ .
Let $S$ be the intersection point of $\omega$ and line $QP$ .
Let the perpendicular bisector of $X_1 X_2$ be line $m$ .
Since $\angle X_1 Q S = \angle X_1 Q P = \angle X_2 X_1 T = \angle X_2 Q T$ ,
points $S$ and $T$ are symmetrical with respect to line $m$ .
also, since $X_1 P=X_2 R$ , points $P$ and $R$ are symmetrical with respect to line $m$ .
Therefore, the two lines $SP$ and $TR$ are symmetrical with respect to line $m$ .

Claim) line $m$ passes through a fixed point, independent of the choice of $l$ .
proof) Let the midpoint of $X_1 X_2$ be $N$ .
It is well known that the trace of $N$ is a circle that passes through $P$ and $Q$ .
Let the trace of $N$ be $\Gamma$ .
Let the antipode of $P$ in $\Gamma$ be $P^*$ .
Then, $\angle P M P^* = 90$ , therefore $M P^*$ is the perpendicular bisector of line $X_1 X_2$ .
Since $P^*$ is fixed, Claim) is proved.

Let $\gamma$ be a circle with center $P^*$ and is tangent to $QP$ .
Since $PS$ and $RT$ is symmetrical WRT $m$ , and the center of $\gamma$ is on $m$ ,
$\gamma$ is also tangent to $RT$ .
Also, $\gamma$ is fixed since the center $P*$ is fixed, and the radius, which is the distance of $PQ$ and $P^*$ , is fixed.
Therefore, $RT$ is tangent to the fixed circle $\gamma$ independent of the choice of $l$ .

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This post has been edited 1 time. Last edited by Han-Ji-Seung, Sep 3, 2022, 4:20 PM
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