Danube Mathematical Competition

The lines $Aa$ and $bC$ meet at $D$, and the lines $Bb$ and $cA$ meet at $D'$ to determine the Pascal line of the hexagram $AaBbCc$; similarly, the lines $Bc$ and $aA$ meet at $E$, and the lines $Ca$ and $bB$ meet at $E'$ to determine the Pascal line of the hexagram $AbBcCa$; finally, the lines $Cb$ and $cB$ meet at $F$, and the lines $Ac$ and $aC$ meet at $F'$ to determine the Pascal line of the hexagram $AcBaCb$. By Desargues' theorem, the lines $DD'$, $EE'$, $FF'$ are concurrent if and only if the pairs of lines $DE$ and $D'E'$, $EF$ and $E'F'$, $FD$ and $F'D'$ meet at three collinear points. Since the latter lie on the Pascal line of the hexagram $AcBbCa$, the conclusion follows.