Stars of Mathematics Competition

We show that the line through the centres of the circles $AMN$ and $AXY$ passes through the centre of the circle $ABC$. Alternatively, but equivalently, we prove that the three circles share a point different from $A$.

Invert the whole configuration from $A$ and let $Z^{\prime }$ denote the image of $Z$ under inversion: The circles $ABC$, $AMN$ and $AXY$ are transformed into the lines $B^{\prime }{C}^{\prime }$, $M^{\prime }{N}^{\prime }$ and $X^{\prime }{Y}^{\prime }$, respectively; the line $BC$ is transformed into the circle $\gamma$ through $A$, $B^{\prime }$ and $C^{\prime }$; the circle ${\gamma }_B$ is transformed into a circle ${\gamma }_B^{\prime }$ through $B^{\prime }$ and $M^{\prime }$, centred at $X^{\prime }$ and externally tangent to $\gamma$ at $B^{\prime }$; and the circle ${\gamma }_C$ is transformed into a circle ${\gamma }_C^{\prime }$ through $C^{\prime }$ and $N^{\prime }$, centred at $Y^{\prime }$ and externally tangent to $\gamma$ at $C^{\prime }$. In this setting, we are to prove that the lines $B^{\prime }{C}^{\prime }$, $M^{\prime }{N}^{\prime }$ and $X^{\prime }{Y}^{\prime }$ are (projectively) concurrent.

To this end, let the pair of external common tangents of ${\gamma }_B^{\prime }$ and ${\gamma }_C^{\prime }$ meet at $O$, and let $\theta$ be the homothety centred at $O$ mapping ${\gamma }_B^{\prime }$ onto ${\gamma }_C^{\prime }$; clearly, $O$ lies on the line $X^{\prime }{Y}^{\prime }$ through the centres of the two circles. Let further ${\theta }_B$ and ${\theta }_C$ be the homotheties centred at $B^{\prime }$ and $C^{\prime }$, respectively, mapping $\gamma$ onto ${\gamma }_B^{\prime }$ and ${\gamma }_C^{\prime }$, respectively.

The centres of the three homotheties are collinear, so $B^{\prime }{C}^{\prime }$ passes through $O$.

Finally, $\theta ={\theta }_C{\theta }_B^{-1}$, so $\theta M^{\prime }={\theta }_C{\theta }_B^{-1}{M}^{\prime }={\theta }_{C}A=N^{\prime }$, showing that the line $M^{\prime }{N}^{\prime }$ passes through $O$ as well. This ends the proof.